%------------------------------------------------------------------------------
% File     : ITP229_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Insert 00617_038319
% 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    : 0066_VEBT_Insert_00617_038319 [Des22]

% Status   : Theorem
% Rating   : 0.89 v8.2.0, 1.00 v8.1.0
% Syntax   : Number of formulae    : 11928 (2840 unt;2466 typ;   0 def)
%            Number of atoms       : 25803 (8538 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 19066 (2725   ~; 497   |;1884   &)
%                                         (2629 <=>;11331  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   15 (   2 avg)
%            Number of types       :  495 ( 494 usr)
%            Number of type conns  : 2274 (1546   >; 728   *;   0   +;   0  <<)
%            Number of predicates  :   67 (  64 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1908 (1908 usr; 426 con; 0-4 aty)
%            Number of variables   : 24742 (24019   !; 723   ?;24742   :)
% SPC      : TF0_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 20:03:59.789
%------------------------------------------------------------------------------
% Could-be-implicit typings (494)
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    fun_rat_num: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Nat__Onat_J,type,
    fun_rat_nat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
    fun_rat_int: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Rat__Orat_J,type,
    fun_num_rat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    fun_num_num: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
    fun_num_nat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    fun_num_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    fun_nat_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
    fun_nat_num: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    fun_nat_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    fun_int_rat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
    fun_int_num: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    fun_int_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    fun_int_int: $tType ).

tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $tType ).

tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
    list_bool: $tType ).

tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

tff(ty_n_t__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_n_t__Set__Oset_It__HOL__Obool_J,type,
    set_bool: $tType ).

tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

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

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

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

tff(ty_n_t__HOL__Obool,type,
    bool: $tType ).

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

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

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

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

% Explicit typings (1972)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_Pr1261947904930325089at_nat > fun_Pr7757163693017731481t_bool ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: real > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_real_bool ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_fun_nat_rat_bool ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_Pr7757163693017731481t_bool > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_Pr7757163693017731481t_bool > fun_fu6897202038601531823t_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_complex > fun_set_complex_bool ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_se1534075524646452397t_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_Pr7757163693017731481t_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_real_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_int_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_int > fun_fu691032689367502339t_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_int > fun_fu8949065344085950851t_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_real_real ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_int_int ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_real_real ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_int_int ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_Pr8693737435421807431at_nat > fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_Pr7459493094073627847at_nat > fun_se5010674451832123565t_bool ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_Pr4329608150637261639at_nat > fun_se1534075524646452397t_bool ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_Pr958786334691620121nt_int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_bool > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_bool_bool > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_bool_nat > fun_fu1660592803699026854at_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_bool > fun_fu737869097161986712t_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_nat_nat > fun_fu1217771548038759377at_nat ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_bool_bool > fun_fu691032689367502339t_bool ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_bo6148171183121378627t_bool > fun_fu5132082026026788591t_bool ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_na4827981024643048934t_bool > fun_fu5855118699552457754t_bool ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_fu5641360819561228055l_bool > fun_fu4959356912570239247t_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_fu4450724535874491238ol_nat > fun_fu7993321427942571930at_nat ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_bo3682073979402284333at_nat > fun_fu5047582607593018841at_nat ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_Pr7757163693017731481t_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_Pr2602117396471556196at_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_Pr248683899324676269t_bool > fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_Pr803113274115125473t_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: bool > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: bool > fun_fu691032689367502339t_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: set_int > fun_fu3139726878529933315t_bool ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: set_int > fun_fu3078129256884107011t_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: set_int > fun_fu2946920718754786320at_int ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: set_nat > fun_fu6165624312919156332nt_int ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_int > fun_fu4844541694308389507t_bool ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_complex > fun_fu8510821397121033091x_bool ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: set_complex > fun_fu8281188428872102787x_bool ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_complex > fun_fu5276431085538847372omplex ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_nat > fun_fu5286008751397070186omplex ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_complex > fun_fu3391973010043818627x_bool ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_complex > fun_fu2147946303232500784omplex ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: set_int > fun_fu2298032621795249330omplex ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: set_nat > fun_fu1296116514128556524l_real ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: set_real > fun_fu4624264860510622736t_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_int_rat > fun_fu2671659548940339672nt_rat ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_complex_rat > fun_fu4947971316162376536ex_rat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_int_nat > fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: produc3843707927480180839at_nat > fun_Pr1607443439640623155t_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_int_fun_int_bool ).

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

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

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

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_Pr1607443439640623155t_bool > fun_Pr124437560471516333t_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_Pr7757163693017731481t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_Pr1607443439640623155t_bool > fun_Pr124437560471516333t_bool ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_Pr7757163693017731481t_bool > fun_Pr248683899324676269t_bool ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_nat > fun_fu7002938529518989751at_nat ).

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

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

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: complex > fun_fu2513798379308225899ex_nat ).

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

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: real > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: int > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: int > fun_fu4328765270694171897t_real ).

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

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_rat > fun_nat_rat ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_complex_nat > fun_complex_complex ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_int_int ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_real_real ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_real > fun_nat_fun_nat_real ).

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

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

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

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

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: complex > fun_co6883864892131739507omplex ).

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

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: rat > fun_ra5513060666784058567at_rat ).

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

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: int > fun_in6793278964266839919at_int ).

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

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: real > fun_na5225841814499866095t_real ).

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

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

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

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

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_real > fun_fu6899955116716589935t_real ).

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

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

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_nat_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_int > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: real > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: complex > fun_fu4213752561377164075ex_int ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: real > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: complex > fun_fu4213752561377164075ex_int ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: set_nat > fun_fu6487107696295728876al_int ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: set_real > fun_fu1164691120062057232at_int ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: set_nat > fun_fu1666499901826068458ex_int ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: set_complex > fun_fu3438847080485518604at_int ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: set_real > fun_fu4833610055523116803l_bool ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: set_real > fun_fu5561217233191876880at_int ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_fun_real_int > fun_real_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: set_nat > fun_fu213344517885440003t_bool ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: set_nat > fun_fu8763428602670138604al_int ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: set_complex > fun_fu3219674935172053772at_int ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_na5967208032043182372ex_int > fun_co7370231399432144932at_int ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: set_nat > fun_fu7887557845401677290ex_int ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: set_nat > fun_fu4745692549347326152at_int ).

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

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: set_nat > fun_fu4745692549347326152at_int ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_real > fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_real_nat > fun_fu4405974721632067664al_nat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_int_nat > fun_fu9051645886418017232nt_nat ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_complex_nat > fun_fu2104585616785278288ex_nat ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_real_int > fun_fu1915891576700340908al_int ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_complex_int > fun_fu8837874508708327340ex_int ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_real > fun_fu9152424228179946668t_real ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_complex_nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_real_int > fun_real_bool ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_complex_int > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: complex > fun_fu6919794090103715845omplex ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: real > fun_fu9096511887113981695omplex ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: int > fun_fu5467039391162264447omplex ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_nat_complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_set_nat_real > fun_set_nat_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_Pr672294278902479040t_real > fun_Pr672294278902479040t_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_nat_fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_fun_nat_nat > fun_nat_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_re8661203372455305201t_bool ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_real_real ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_nat_real > fun_nat_real ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: fun_rat_rat ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: fun_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: fun_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: fun_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: fun_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,
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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: fun_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: fun_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 ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_bool ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option_num ).

tff(sy_c_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    cOMBB_3745307990312268622T_VEBT: ( fun_bool_bool * fun_VEBT_VEBT_bool ) > fun_VEBT_VEBT_bool ).

tff(sy_c_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT,type,
    cOMBB_3438363824644082366T_VEBT: ( fun_bo1841576699660343469l_bool * fun_VEBT_VEBT_bool ) > fun_VE2066785035971159626l_bool ).

tff(sy_c_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    cOMBB_2885587777813823575T_VEBT: ( fun_fu7168253649841883962l_bool * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_bool ).

tff(sy_c_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool,type,
    cOMBC_771164880915498576T_bool: ( fun_VE7346385682144900493T_bool * set_VEBT_VEBT ) > fun_VEBT_VEBT_bool ).

tff(sy_c_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool,type,
    cOMBS_6222329278419335549l_bool: ( fun_VE2066785035971159626l_bool * fun_VEBT_VEBT_bool ) > fun_VEBT_VEBT_bool ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: fun_set_nat_nat ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: fun_set_nat_nat ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

tff(sy_c_Complex_OArg,type,
    arg: fun_complex_real ).

tff(sy_c_Complex_Ocis,type,
    cis: fun_real_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > fun_real_complex ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
    condit2214826472909112428ve_nat: fun_set_nat_bool ).

tff(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( fun_real_real * filter_real ) > $o ).

tff(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
    has_de1759254742604945161l_real: ( fun_real_real * fun_real_real * filter_real ) > $o ).

tff(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( fun_real_real * real * filter_real ) > $o ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int ) > fun_Pr803113274115125473t_bool ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: fun_Pr803113274115125473t_bool ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: fun_Pr7757163693017731481t_bool ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: ( num * num ) > product_prod_int_int ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: ( num * num ) > product_prod_nat_nat ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: ( num * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
    unique5026877609467782581ep_nat: ( num * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > fun_nat_complex ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
    comm_s4660882817536571857er_int: int > fun_nat_int ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
    comm_s4663373288045622133er_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
    comm_s4028243227959126397er_rat: rat > fun_nat_rat ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
    comm_s7457072308508201937r_real: fun_re4238438383274954274t_real ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: fun_nat_nat ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: fun_nat_real ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: fun_real_real ).

tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

tff(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

tff(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

tff(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( fun_nat_bool * filter_nat ) > $o ).

tff(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    eventu1038000079068216329at_nat: ( fun_Pr7757163693017731481t_bool * filter1242075044329608583at_nat ) > $o ).

tff(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
    eventually_real: ( fun_real_bool * filter_real ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( fun_nat_nat * filter_nat * filter_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( fun_nat_real * filter_real * filter_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
    filterlim_real_real: ( fun_real_real * filter_real * filter_real ) > $o ).

tff(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).

tff(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).

tff(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
    prod_filter_nat_nat: ( filter_nat * filter_nat ) > filter1242075044329608583at_nat ).

tff(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
    linord738340561235409698at_nat: ( fun_nat_nat * list_nat ) > list_nat ).

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

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

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

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

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

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

tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
    set_bool2: list_bool > set_bool ).

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

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

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

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

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

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

tff(sy_c_List_Olist__update_001t__HOL__Obool,type,
    list_update_bool: ( list_bool * nat * bool ) > list_bool ).

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

tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: ( list_real * nat * real ) > list_real ).

tff(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Onth_001t__HOL__Obool,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__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
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tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: ( list_set_nat * nat ) > set_nat ).

tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__HOL__Obool,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__HOL__Obool,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_Oreplicate_001t__HOL__Obool,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: ( nat * int ) > list_int ).

tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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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,
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tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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tff(sy_c_List_Otake_001t__Nat__Onat,type,
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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,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
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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,
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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,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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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__HOL__Obool_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat__Bijection_Oint__decode,type,
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tff(sy_c_Nat__Bijection_Oint__encode,type,
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tff(sy_c_Nat__Bijection_Olist__encode,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__encode,type,
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    nat_set_decode: nat > set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

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

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_ma7524802468073614006at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

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

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

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
    power_8040749407984259932d_enat: ( extended_enat * nat ) > extended_enat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__HOL__Obool,type,
    produc8297744024737662807l_bool: ( bool * bool ) > produc2804947525362035943l_bool ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
    produc8871513391732212460ol_int: ( bool * int ) > produc5612625445546783218ol_int ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Nat__Onat,type,
    produc8874003862241262736ol_nat: ( bool * nat ) > produc567104428201204118ol_nat ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    produc9074556814250467700T_VEBT: ( bool * vEBT_VEBT ) > produc8408751633324799108T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: ( int * int ) > product_prod_int_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__HOL__Obool,type,
    produc3913204875133826030t_bool: ( nat * bool ) > produc8000372497851584572t_bool ).

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: ( nat * product_prod_nat_nat ) > produc7248412053542808358at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: ( product_prod_nat_nat * product_prod_nat_nat ) > produc859450856879609959at_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2922128104949294807at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > produc3843707927480180839at_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    produc9060074326276436823at_nat: ( set_Pr4329608150637261639at_nat * set_Pr4329608150637261639at_nat ) > produc1319942482725812455at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: ( vEBT_VEBT * nat ) > produc9072475918466114483BT_nat ).

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2818902965508115497at_nat: ( fun_nat_fun_nat_nat * produc7248412053542808358at_nat ) > produc9217583074051984497at_nat ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool,type,
    produc8560705615207772305t_bool: fun_int_fun_int_bool > fun_Pr803113274115125473t_bool ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    produc73460835934605544et_int: ( fun_in7040156458970095540et_int * product_prod_int_int ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4251311855443802252et_nat: ( fun_in2630332840903238488et_nat * product_prod_int_int ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc1656060378719767003at_nat: ( fun_in1904380326158164027at_nat * product_prod_int_int ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
    produc6452406959799940328t_real: ( fun_in6553099857819052212t_real * product_prod_int_int ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Rat_OFract,type,
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tff(sy_c_Rat_ORep__Rat,type,
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tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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tff(sy_c_Rat_Onormalize,type,
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tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_bool ).

tff(sy_c_Real_OReal,type,
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tff(sy_c_Real_Ocauchy,type,
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tff(sy_c_Real_Opositive,type,
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tff(sy_c_Real_Orep__real,type,
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tff(sy_c_Real_Ovanishes,type,
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tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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tff(sy_c_Relation_OField_001t__Nat__Onat,type,
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tff(sy_c_Relation_OId_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
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tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__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_001t__Complex__Ocomplex,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__HOL__Obool_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__Nat__Onat_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__Num__Onum,type,
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tff(sy_c_Set_OCollect_001t__Rat__Orat,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__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
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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__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__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__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
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tff(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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tff(sy_c_Set_Oinsert_001t__Num__Onum,type,
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    insert9069300056098147895at_nat: ( produc3843707927480180839at_nat * set_Pr4329608150637261639at_nat ) > set_Pr4329608150637261639at_nat ).

tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
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tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: ( set_nat * set_set_nat ) > set_set_nat ).

tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
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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__Rat__Orat,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_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or370866239135849197et_int: ( set_int * set_int ) > set_set_int ).

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

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: fun_in7040156458970095540et_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: fun_na6128946301132549280et_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

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

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

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,
    set_or6656581121297822940st_int: fun_in7040156458970095540et_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: fun_na6128946301132549280et_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: fun_in7040156458970095540et_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: fun_na6128946301132549280et_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

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

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

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

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

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

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

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > 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_VE3763956218935105199t_bool ).

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_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

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

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_bool ).

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_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
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tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
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tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
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tff(sy_c_VEBT__Member_Ovebt__member,type,
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tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
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tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Oless__than,type,
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tff(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
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tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Rat__Orat,type,
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tff(sy_c_aa_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Int__Oint_001t__Num__Onum,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_It__Int__Oint_J,type,
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    member8757157785044589968at_nat: produc3843707927480180839at_nat > fun_se8353504738346895699t_bool ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member1466754251312161552at_nat: ( produc1319942482725812455at_nat * set_Pr7459493094073627847at_nat ) > $o ).

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

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

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

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

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_ya____,type,
    ya: nat ).

% Relevant facts (9401)
tff(fact_0_False,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),ma)) ).

% False
tff(fact_1__092_060open_062x_A_060_Ami_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)) ).

% \<open>x < mi\<close>
tff(fact_2__C4_Ohyps_C_I7_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% "4.hyps"(7)
tff(fact_3__C4_Oprems_C_I3_J,axiom,
    pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa)),ya)) ).

% "4.prems"(3)
tff(fact_4_True,axiom,
    ( ( ya = xa )
    | ( ya = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,mi),ma) ) ) ).

% True
tff(fact_5__092_060open_062y_A_061_Amax_Ami_Ama_092_060close_062,axiom,
    ya = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,mi),ma) ).

% \<open>y = max mi ma\<close>
tff(fact_6_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_7_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_8_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_9_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_10_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( produc6161850002892822231at_nat(X1,X2) = produc6161850002892822231at_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_11_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( ( produc9060074326276436823at_nat(X1,X2) = produc9060074326276436823at_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_12_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(X1,X2) = produc2922128104949294807at_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

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

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

% prod.inject
tff(fact_15_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( produc6161850002892822231at_nat(A,B) = produc6161850002892822231at_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_16_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( produc9060074326276436823at_nat(A,B) = produc9060074326276436823at_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_17_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(A,B) = produc2922128104949294807at_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

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

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

% old.prod.inject
tff(fact_20__C4_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X3: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) ) ).

% "4.hyps"(6)
tff(fact_21_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),M) ).

% prod_decode_aux.cases
tff(fact_22__C4_Ohyps_C_I1_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).

% "4.hyps"(1)
tff(fact_23_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : Y != produc6161850002892822231at_nat(A3,B3) ).

% old.prod.exhaust
tff(fact_24_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1319942482725812455at_nat] :
      ~ ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : Y != produc9060074326276436823at_nat(A3,B3) ).

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

% old.prod.exhaust
tff(fact_26_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) ).

% old.prod.exhaust
tff(fact_27_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y != product_Pair_int_int(A3,B3) ).

% old.prod.exhaust
tff(fact_28_surj__pair,axiom,
    ! [P: produc859450856879609959at_nat] :
    ? [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] : P = produc6161850002892822231at_nat(X4,Y3) ).

% surj_pair
tff(fact_29_surj__pair,axiom,
    ! [P: produc1319942482725812455at_nat] :
    ? [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : P = produc9060074326276436823at_nat(X4,Y3) ).

% surj_pair
tff(fact_30_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : P = produc2922128104949294807at_nat(X4,Y3) ).

% surj_pair
tff(fact_31_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X4: nat,Y3: nat] : P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) ).

% surj_pair
tff(fact_32_surj__pair,axiom,
    ! [P: product_prod_int_int] :
    ? [X4: int,Y3: int] : P = product_Pair_int_int(X4,Y3) ).

% surj_pair
tff(fact_33_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_34_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12)) ) ).

% not_min_Null_member
tff(fact_35_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_36_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_37_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_38_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) )
          & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_39_prod__induct3,axiom,
    ! [P2: fun_Pr446604468264085555t_bool,X: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: nat,C: nat] : pp(aa_Pro6098352494050694700t_bool(P2,produc6161850002892822231at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),C))))
     => pp(aa_Pro6098352494050694700t_bool(P2,X)) ) ).

% prod_induct3
tff(fact_40_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: nat,C: nat] : Y != produc6161850002892822231at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),C)) ).

% prod_cases3
tff(fact_41_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( produc6161850002892822231at_nat(A,B) = produc6161850002892822231at_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_42_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( produc9060074326276436823at_nat(A,B) = produc9060074326276436823at_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_43_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(A,B) = produc2922128104949294807at_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_44_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

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

% Pair_inject
tff(fact_46_prod__cases,axiom,
    ! [P2: fun_Pr446604468264085555t_bool,P: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : pp(aa_Pro6098352494050694700t_bool(P2,produc6161850002892822231at_nat(A3,B3)))
     => pp(aa_Pro6098352494050694700t_bool(P2,P)) ) ).

% prod_cases
tff(fact_47_prod__cases,axiom,
    ! [P2: fun_Pr2985331876022323379t_bool,P: produc1319942482725812455at_nat] :
      ( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : pp(aa_Pro4324259192382070444t_bool(P2,produc9060074326276436823at_nat(A3,B3)))
     => pp(aa_Pro4324259192382070444t_bool(P2,P)) ) ).

% prod_cases
tff(fact_48_prod__cases,axiom,
    ! [P2: fun_Pr1607443439640623155t_bool,P: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : pp(aa_Pro1809015098595223084t_bool(P2,produc2922128104949294807at_nat(A3,B3)))
     => pp(aa_Pro1809015098595223084t_bool(P2,P)) ) ).

% prod_cases
tff(fact_49_prod__cases,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,P: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : pp(aa_Pro2232559814647311250t_bool(P2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3)))
     => pp(aa_Pro2232559814647311250t_bool(P2,P)) ) ).

% prod_cases
tff(fact_50_prod__cases,axiom,
    ! [P2: fun_Pr803113274115125473t_bool,P: product_prod_int_int] :
      ( ! [A3: int,B3: int] : pp(aa_Pro4501881432599481050t_bool(P2,product_Pair_int_int(A3,B3)))
     => pp(aa_Pro4501881432599481050t_bool(P2,P)) ) ).

% prod_cases
tff(fact_51__C4_Ohyps_C_I3_J,axiom,
    m = na ).

% "4.hyps"(3)
tff(fact_52_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_53_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_54_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb3
tff(fact_55_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_56_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_57_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_58_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_59_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb4
tff(fact_60_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_61_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_62_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_63_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(X),Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_64_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(X),Y)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_65_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_66_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_67_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_68_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb1
tff(fact_69_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_70_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_71_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_72_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb2
tff(fact_73_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_74_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_75_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_76_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_77_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_78_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_79_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_80_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_81_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P2: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),collec3392354462482085612at_nat(P2)))
    <=> pp(aa_Pro2232559814647311250t_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_82_mem__Collect__eq,axiom,
    ! [A: real,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A),collect_real(P2)))
    <=> pp(aa_real_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_83_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P2))
    <=> pp(aa_list_nat_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_84_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),collect_set_nat(P2)))
    <=> pp(aa_set_nat_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_85_mem__Collect__eq,axiom,
    ! [A: nat,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P2)))
    <=> pp(aa_nat_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_86_mem__Collect__eq,axiom,
    ! [A: int,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P2)))
    <=> pp(aa_int_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_87_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A4)) = A4 ).

% Collect_mem_eq
tff(fact_88_Collect__mem__eq,axiom,
    ! [A4: set_real] : collect_real(aTP_Lamm_aa(A4)) = A4 ).

% Collect_mem_eq
tff(fact_89_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A4)) = A4 ).

% Collect_mem_eq
tff(fact_90_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A4)) = A4 ).

% Collect_mem_eq
tff(fact_91_Collect__mem__eq,axiom,
    ! [A4: set_nat] : collect_nat(aTP_Lamm_ad(A4)) = A4 ).

% Collect_mem_eq
tff(fact_92_Collect__mem__eq,axiom,
    ! [A4: set_int] : collect_int(aTP_Lamm_ae(A4)) = A4 ).

% Collect_mem_eq
tff(fact_93_Collect__cong,axiom,
    ! [P2: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P2,X4))
        <=> pp(aa_real_bool(Q,X4)) )
     => ( collect_real(P2) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_94_Collect__cong,axiom,
    ! [P2: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P2,X4))
        <=> pp(aa_list_nat_bool(Q,X4)) )
     => ( collect_list_nat(P2) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_95_Collect__cong,axiom,
    ! [P2: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P2,X4))
        <=> pp(aa_set_nat_bool(Q,X4)) )
     => ( collect_set_nat(P2) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_96_Collect__cong,axiom,
    ! [P2: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P2,X4))
        <=> pp(aa_nat_bool(Q,X4)) )
     => ( collect_nat(P2) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_97_Collect__cong,axiom,
    ! [P2: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P2,X4))
        <=> pp(aa_int_bool(Q,X4)) )
     => ( collect_int(P2) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_98_max_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),A) = A ).

% max.idem
tff(fact_99_max_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),A) = A ).

% max.idem
tff(fact_100_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_101_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_102_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_103_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_104__C4_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "4.hyps"(4)
tff(fact_105_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: 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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)) ).

% max.left_commute
tff(fact_106_max_Oleft__commute,axiom,
    ! [B: int,A: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)) ).

% max.left_commute
tff(fact_107_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_108_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_109_max_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)) ).

% max.assoc
tff(fact_110_max_Oassoc,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)) ).

% max.assoc
tff(fact_111_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI2
tff(fact_112_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),aa_num_num(ord_max_num(A),B))) ) ).

% max.coboundedI2
tff(fact_113_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI2
tff(fact_114_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI2
tff(fact_115_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI1
tff(fact_116_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),aa_num_num(ord_max_num(A),B))) ) ).

% max.coboundedI1
tff(fact_117_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI1
tff(fact_118_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI1
tff(fact_119_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_120_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_121_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_122_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_123_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_124_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_125_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_126_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_127_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_128_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),aa_num_num(ord_max_num(X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_129_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_130_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_131_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded2
tff(fact_132_max_Ocobounded2,axiom,
    ! [B: num,A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),aa_num_num(ord_max_num(A),B))) ).

% max.cobounded2
tff(fact_133_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded2
tff(fact_134_max_Ocobounded2,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded2
tff(fact_135_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded1
tff(fact_136_max_Ocobounded1,axiom,
    ! [A: num,B: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(ord_max_num(A),B))) ).

% max.cobounded1
tff(fact_137_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded1
tff(fact_138_max_Ocobounded1,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded1
tff(fact_139_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_140_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.order_iff
tff(fact_141_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_142_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_143_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A)) ) ) ).

% max.boundedI
tff(fact_144_max_OboundedI,axiom,
    ! [B: num,A: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A)) ) ) ).

% max.boundedI
tff(fact_145_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)) ) ) ).

% max.boundedI
tff(fact_146_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)) ) ) ).

% max.boundedI
tff(fact_147_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A)) ) ) ).

% max.boundedE
tff(fact_148_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A)) ) ) ).

% max.boundedE
tff(fact_149_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A)) ) ) ).

% max.boundedE
tff(fact_150_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A)) ) ) ).

% max.boundedE
tff(fact_151_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% max.orderI
tff(fact_152_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(ord_max_num(A),B) )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% max.orderI
tff(fact_153_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% max.orderI
tff(fact_154_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% max.orderI
tff(fact_155_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_156_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.orderE
tff(fact_157_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_158_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_159_max_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C2),D)),aa_rat_rat(ord_max_rat(A),B))) ) ) ).

% max.mono
tff(fact_160_max_Omono,axiom,
    ! [C2: num,A: num,D: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,D),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(C2),D)),aa_num_num(ord_max_num(A),B))) ) ) ).

% max.mono
tff(fact_161_max_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).

% max.mono
tff(fact_162_max_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C2),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ) ).

% max.mono
tff(fact_163_max_Ostrict__coboundedI2,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_164_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_165_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),aa_num_num(ord_max_num(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_166_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_167_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_168_max_Ostrict__coboundedI1,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_169_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_170_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),aa_num_num(ord_max_num(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_171_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_172_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_173_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_174_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_175_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( ( A = aa_num_num(ord_max_num(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_176_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_177_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_178_max_Ostrict__boundedE,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C2)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A)) ) ) ).

% max.strict_boundedE
tff(fact_179_max_Ostrict__boundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(B),C2)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A)) ) ) ).

% max.strict_boundedE
tff(fact_180_max_Ostrict__boundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(B),C2)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A)) ) ) ).

% max.strict_boundedE
tff(fact_181_max_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A)) ) ) ).

% max.strict_boundedE
tff(fact_182_max_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A)) ) ) ).

% max.strict_boundedE
tff(fact_183_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_184_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(ord_max_rat(X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_185_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(ord_max_num(X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_186_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_187_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_188_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
     => pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq2
tff(fact_189_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),D))
     => pp(aa_nat_bool(vEBT_VEBT_valid(T),D)) ) ).

% valid_eq1
tff(fact_190_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
    <=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq
tff(fact_191_dual__order_Orefl,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% dual_order.refl
tff(fact_192_dual__order_Orefl,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% dual_order.refl
tff(fact_193_dual__order_Orefl,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% dual_order.refl
tff(fact_194_dual__order_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% dual_order.refl
tff(fact_195_dual__order_Orefl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% dual_order.refl
tff(fact_196_order__refl,axiom,
    ! [X: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X),X)) ).

% order_refl
tff(fact_197_order__refl,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),X)) ).

% order_refl
tff(fact_198_order__refl,axiom,
    ! [X: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),X)) ).

% order_refl
tff(fact_199_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_200_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_201_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_202_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_203_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( ord_max_set_int(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_204_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(ord_max_rat(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_205_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( aa_num_num(ord_max_num(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_206_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_207_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_208_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( ord_max_set_int(X,Y) = X ) ) ).

% max_absorb1
tff(fact_209_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(ord_max_rat(X),Y) = X ) ) ).

% max_absorb1
tff(fact_210_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( aa_num_num(ord_max_num(X),Y) = X ) ) ).

% max_absorb1
tff(fact_211_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_212_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_213_max__def,axiom,
    ! [A: set_int,B: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( ord_max_set_int(A,B) = B ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( ord_max_set_int(A,B) = A ) ) ) ).

% max_def
tff(fact_214_max__def,axiom,
    ! [A: rat,B: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = B ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ) ).

% max_def
tff(fact_215_max__def,axiom,
    ! [A: num,B: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(ord_max_num(A),B) = B ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(ord_max_num(A),B) = A ) ) ) ).

% max_def
tff(fact_216_max__def,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ) ).

% max_def
tff(fact_217_max__def,axiom,
    ! [A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ) ).

% max_def
tff(fact_218_nat__descend__induct,axiom,
    ! [N: nat,P2: fun_nat_bool,M2: nat] :
      ( ! [K: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
         => pp(aa_nat_bool(P2,K)) )
     => ( ! [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
           => ( ! [I: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),I))
                 => pp(aa_nat_bool(P2,I)) )
             => pp(aa_nat_bool(P2,K)) ) )
       => pp(aa_nat_bool(P2,M2)) ) ) ).

% nat_descend_induct
tff(fact_219_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_220_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_221_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S),X) = vEBT_Node(Info,zero_zero_nat,Ts,S) ).

% vebt_insert.simps(2)
tff(fact_222_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_223_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_224_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_225_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_226_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) )
     => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z))
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) )
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y))
             => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) )
           => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
               => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_227_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
       => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z)) )
     => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z))
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) )
         => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y))
             => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) )
           => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
               => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_228_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_229_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_230_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_231_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_232_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_233_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_234_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_235_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( A = B )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_236_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_237_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_238_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_239_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_240_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( ( B = C2 )
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_241_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( B = C2 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_242_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( B = C2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_243_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( B = C2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_244_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( B = C2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_245_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_246_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_247_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_248_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_249_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_250_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C2)) ) ) ).

% order.trans
tff(fact_251_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2)) ) ) ).

% order.trans
tff(fact_252_order_Otrans,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C2)) ) ) ).

% order.trans
tff(fact_253_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2)) ) ) ).

% order.trans
tff(fact_254_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2)) ) ) ).

% order.trans
tff(fact_255_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X),Z)) ) ) ).

% order_trans
tff(fact_256_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) ) ) ).

% order_trans
tff(fact_257_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) ) ) ).

% order_trans
tff(fact_258_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).

% order_trans
tff(fact_259_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) ) ) ).

% order_trans
tff(fact_260_linorder__wlog,axiom,
    ! [P2: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A3),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),B3)) )
     => ( ! [A3: rat,B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,B3),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),B3)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_261_linorder__wlog,axiom,
    ! [P2: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),B3)) )
     => ( ! [A3: num,B3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P2,B3),A3))
           => pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),B3)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_262_linorder__wlog,axiom,
    ! [P2: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3)) )
     => ( ! [A3: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,B3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_263_linorder__wlog,axiom,
    ! [P2: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),B3)) )
     => ( ! [A3: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P2,B3),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_264_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_265_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_266_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_267_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_268_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_269_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_270_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_271_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_272_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_273_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_274_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),B))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C2),A)) ) ) ).

% dual_order.trans
tff(fact_275_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A)) ) ) ).

% dual_order.trans
tff(fact_276_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A)) ) ) ).

% dual_order.trans
tff(fact_277_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A)) ) ) ).

% dual_order.trans
tff(fact_278_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A)) ) ) ).

% dual_order.trans
tff(fact_279_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_280_antisym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_281_antisym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_282_antisym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_283_antisym,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_284_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_285_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_286_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_287_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_288_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_289_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_290_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_291_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_292_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_293_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_294_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_295_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_296_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,C2))) ) ) ) ).

% order_subst1
tff(fact_297_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_298_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C2))) ) ) ) ).

% order_subst1
tff(fact_299_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_300_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_301_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_302_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_303_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_304_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_305_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_306_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_307_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_308_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B)),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_309_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_eq_refl
tff(fact_310_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_eq_refl
tff(fact_311_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_eq_refl
tff(fact_312_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_313_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_eq_refl
tff(fact_314_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_linear
tff(fact_315_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_linear
tff(fact_316_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_linear
tff(fact_317_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_linear
tff(fact_318_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_319_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_320_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_321_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_322_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_323_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( ( A = aa_num_num(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_324_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_325_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C2: num] :
      ( ( A = aa_num_int(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_326_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_327_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C2: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_328_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_329_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_num(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_330_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_331_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_int(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_332_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_rat(F,B) = C2 )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_333_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_num(F,B) = C2 )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_334_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_nat(F,B) = C2 )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_335_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_int(F,B) = C2 )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_336_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_rat(F,B) = C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_337_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_num(F,B) = C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_338_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_le_cases
tff(fact_339_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_le_cases
tff(fact_340_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_le_cases
tff(fact_341_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_le_cases
tff(fact_342_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_343_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_344_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_345_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_346_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_347_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X)) ).

% lt_ex
tff(fact_348_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X)) ).

% lt_ex
tff(fact_349_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X)) ).

% lt_ex
tff(fact_350_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_12)) ).

% gt_ex
tff(fact_351_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_12)) ).

% gt_ex
tff(fact_352_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_12)) ).

% gt_ex
tff(fact_353_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_12)) ).

% gt_ex
tff(fact_354_dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),Y)) ) ) ).

% dense
tff(fact_355_dense,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ? [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),Y)) ) ) ).

% dense
tff(fact_356_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_357_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_358_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_359_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_360_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_361_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order.asym
tff(fact_362_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order.asym
tff(fact_363_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order.asym
tff(fact_364_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order.asym
tff(fact_365_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order.asym
tff(fact_366_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_367_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_368_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_369_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_370_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_371_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C2 )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_372_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C2 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_373_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_374_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_375_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_376_less__induct,axiom,
    ! [P2: fun_nat_bool,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X4))
             => pp(aa_nat_bool(P2,Y4)) )
         => pp(aa_nat_bool(P2,X4)) )
     => pp(aa_nat_bool(P2,A)) ) ).

% less_induct
tff(fact_377_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_378_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_379_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_380_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_381_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_382_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ( X != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_cases
tff(fact_383_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ( X != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_cases
tff(fact_384_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( ( X != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_cases
tff(fact_385_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_386_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_387_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% dual_order.asym
tff(fact_388_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% dual_order.asym
tff(fact_389_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% dual_order.asym
tff(fact_390_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% dual_order.asym
tff(fact_391_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% dual_order.asym
tff(fact_392_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_393_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% dual_order.irrefl
tff(fact_394_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% dual_order.irrefl
tff(fact_395_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% dual_order.irrefl
tff(fact_396_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% dual_order.irrefl
tff(fact_397_exists__least__iff,axiom,
    ! [P2: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P2,X_13))
    <=> ? [N2: nat] :
          ( pp(aa_nat_bool(P2,N2))
          & ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
             => ~ pp(aa_nat_bool(P2,M3)) ) ) ) ).

% exists_least_iff
tff(fact_398_linorder__less__wlog,axiom,
    ! [P2: fun_re1255401033344146673l_bool,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),B3)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),A3))
       => ( ! [A3: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,B3),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_399_linorder__less__wlog,axiom,
    ! [P2: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),B3)) )
     => ( ! [A3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),A3))
       => ( ! [A3: rat,B3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,B3),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),B3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_400_linorder__less__wlog,axiom,
    ! [P2: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),B3)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),A3))
       => ( ! [A3: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P2,B3),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_401_linorder__less__wlog,axiom,
    ! [P2: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_402_linorder__less__wlog,axiom,
    ! [P2: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P2,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_403_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% order.strict_trans
tff(fact_404_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% order.strict_trans
tff(fact_405_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% order.strict_trans
tff(fact_406_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% order.strict_trans
tff(fact_407_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% order.strict_trans
tff(fact_408_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_409_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_410_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_411_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_412_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_413_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_414_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_415_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_416_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_417_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_418_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_419_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_420_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_421_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_422_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_423_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_424_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_425_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_426_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_427_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_428_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE
tff(fact_429_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_430_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neqE
tff(fact_431_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_432_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE
tff(fact_433_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_asym
tff(fact_434_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_asym
tff(fact_435_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_asym
tff(fact_436_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_asym
tff(fact_437_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_asym
tff(fact_438_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_439_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_440_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_441_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_442_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_443_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order_less_asym'
tff(fact_444_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order_less_asym'
tff(fact_445_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order_less_asym'
tff(fact_446_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order_less_asym'
tff(fact_447_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order_less_asym'
tff(fact_448_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_trans
tff(fact_449_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_trans
tff(fact_450_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_trans
tff(fact_451_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_trans
tff(fact_452_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_trans
tff(fact_453_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( ( A = aa_real_real(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_454_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_455_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C2: real] :
      ( ( A = aa_real_num(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_456_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C2: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_457_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C2: real] :
      ( ( A = aa_real_int(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_458_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_459_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_460_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_461_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_462_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_463_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,B) = C2 )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_464_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F,B) = C2 )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_465_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F,B) = C2 )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_466_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F,B) = C2 )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_467_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F,B) = C2 )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_468_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_469_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_470_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_471_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_472_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F,B) = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_473_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_474_order__less__irrefl,axiom,
    ! [X: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X)) ).

% order_less_irrefl
tff(fact_475_order__less__irrefl,axiom,
    ! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).

% order_less_irrefl
tff(fact_476_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_477_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_478_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_479_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_480_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_481_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_482_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_483_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_484_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_485_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_486_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_487_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C2))) ) ) ) ).

% order_less_subst1
tff(fact_488_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_489_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B)),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_490_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B)),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_491_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B)),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_492_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B)),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_493_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_494_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_495_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_496_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_497_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_498_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_not_sym
tff(fact_499_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_not_sym
tff(fact_500_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_not_sym
tff(fact_501_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_not_sym
tff(fact_502_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_not_sym
tff(fact_503_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P2: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => pp(P2) ) ) ).

% order_less_imp_triv
tff(fact_504_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P2: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
       => pp(P2) ) ) ).

% order_less_imp_triv
tff(fact_505_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P2: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
       => pp(P2) ) ) ).

% order_less_imp_triv
tff(fact_506_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
       => pp(P2) ) ) ).

% order_less_imp_triv
tff(fact_507_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P2: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
       => pp(P2) ) ) ).

% order_less_imp_triv
tff(fact_508_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      | ( X = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_less_linear
tff(fact_509_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      | ( X = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_less_linear
tff(fact_510_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      | ( X = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_less_linear
tff(fact_511_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_512_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_513_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_514_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_515_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_516_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_517_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_518_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_519_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_520_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_521_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_522_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_523_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_524_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_525_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_526_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_527_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_528_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_529_leD,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% leD
tff(fact_530_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X))
     => ~ pp(aa_set_int_bool(ord_less_set_int(X),Y)) ) ).

% leD
tff(fact_531_leD,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% leD
tff(fact_532_leD,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% leD
tff(fact_533_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_534_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_535_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% leI
tff(fact_536_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% leI
tff(fact_537_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% leI
tff(fact_538_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% leI
tff(fact_539_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% leI
tff(fact_540_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_541_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_542_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_543_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_544_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_545_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_546_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_547_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_548_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_549_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_550_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_551_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_552_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_553_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( ~ pp(aa_set_int_bool(ord_less_set_int(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_554_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_555_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_556_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_557_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_558_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_ge
tff(fact_559_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_ge
tff(fact_560_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_le
tff(fact_561_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Z)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_le
tff(fact_562_less__le__not__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% less_le_not_le
tff(fact_563_less__le__not__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ) ).

% less_le_not_le
tff(fact_564_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_565_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% less_le_not_le
tff(fact_566_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_567_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_568_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% not_le_imp_less
tff(fact_569_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% not_le_imp_less
tff(fact_570_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% not_le_imp_less
tff(fact_571_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_572_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_573_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_574_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_575_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_576_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_577_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_578_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_579_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_580_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_581_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_582_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_583_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_584_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_585_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_586_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_set_int(B),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A),C2)) ) ) ).

% order.strict_trans1
tff(fact_587_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_588_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_589_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_590_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_591_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_592_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A),C2)) ) ) ).

% order.strict_trans2
tff(fact_593_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_594_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_595_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_596_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_597_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_598_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% order.strict_iff_not
tff(fact_599_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_600_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_601_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_602_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_603_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),X))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_604_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),X))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_605_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_606_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Y))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_607_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_608_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_609_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_610_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_611_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_612_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_613_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_614_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_615_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_616_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_617_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_618_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_619_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_620_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_set_int(C2),B))
       => pp(aa_set_int_bool(ord_less_set_int(C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_621_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_622_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_623_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_624_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_625_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_626_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),B))
       => pp(aa_set_int_bool(ord_less_set_int(C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_627_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_628_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_629_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_630_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_631_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_632_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_633_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_634_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_635_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_636_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_637_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% order.strict_implies_order
tff(fact_638_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% order.strict_implies_order
tff(fact_639_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% order.strict_implies_order
tff(fact_640_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% order.strict_implies_order
tff(fact_641_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% order.strict_implies_order
tff(fact_642_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% order.strict_implies_order
tff(fact_643_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_644_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
     => pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_645_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_646_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_647_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_648_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_649_order__le__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_650_order__le__less,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_651_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_652_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_653_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_654_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_655_order__less__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_656_order__less__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_657_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_658_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_659_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_660_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_661_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_not_le
tff(fact_662_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_not_le
tff(fact_663_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_not_le
tff(fact_664_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_665_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_666_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_not_less
tff(fact_667_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_not_less
tff(fact_668_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_not_less
tff(fact_669_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_670_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_671_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% order_less_imp_le
tff(fact_672_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% order_less_imp_le
tff(fact_673_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_less_imp_le
tff(fact_674_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_less_imp_le
tff(fact_675_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_676_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_677_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ( A != B )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_678_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( ( A != B )
       => pp(aa_set_int_bool(ord_less_set_int(A),B)) ) ) ).

% order_le_neq_trans
tff(fact_679_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( A != B )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_680_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( A != B )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_681_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( A != B )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_682_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( A != B )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_683_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_684_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => pp(aa_set_int_bool(ord_less_set_int(A),B)) ) ) ).

% order_neq_le_trans
tff(fact_685_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_686_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_687_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_688_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_689_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_690_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z)) ) ) ).

% order_le_less_trans
tff(fact_691_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_692_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_693_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_694_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_695_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_696_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X),Z)) ) ) ).

% order_less_le_trans
tff(fact_697_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_698_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_699_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_700_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_701_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_702_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_703_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_704_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_705_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_706_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_707_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_708_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_709_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_710_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_711_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_712_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_713_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_714_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_715_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_716_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C2: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_717_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_718_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_719_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_720_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_721_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_722_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_723_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_724_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_725_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_726_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_727_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_728_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_729_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_730_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_731_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_732_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_733_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C2: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_734_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B)),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_735_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C2: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B)),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_736_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B)),C2))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_737_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C2))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_738_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C2))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_739_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C2))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_740_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B)),C2))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_741_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_742_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_743_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_744_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_745_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_746_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_747_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_748_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_749_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_750_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_751_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_752_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_753_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_754_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_755_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_756_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_757_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_758_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_759_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_760_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_761_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_762_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel1
tff(fact_763_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel1
tff(fact_764_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel1
tff(fact_765_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel1
tff(fact_766_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel2
tff(fact_767_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel2
tff(fact_768_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel2
tff(fact_769_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel2
tff(fact_770_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_771_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_772_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_773_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_774_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_775_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_776_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_777_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_778_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_779_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_780_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_781_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_782_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_783_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_784_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel1
tff(fact_785_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel1
tff(fact_786_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel1
tff(fact_787_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel1
tff(fact_788_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel2
tff(fact_789_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel2
tff(fact_790_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel2
tff(fact_791_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel2
tff(fact_792_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_793_add__right__cancel,axiom,
    ! [B: int,A: int,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_794_add__right__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_795_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_796_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_797_add__left__cancel,axiom,
    ! [A: int,B: int,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_798_add__left__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_799_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_800_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_801_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_802_add__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_803_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_804_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_805_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_806_add__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_807_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_808_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_809_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_810_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_811_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_812_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_813_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_814_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_815_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_816_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_817_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_818_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_819_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_820_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_821_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_822_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_823_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_824_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_825_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_826_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_827_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_828_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_829_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_830_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_831_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_832_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_833_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_834_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_835_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_836_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_837_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_838_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_839_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_840_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_841_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_842_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_843_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_844_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_845_add__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_846_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_847_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_848_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_849_add__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_850_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_851_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_852_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_853_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_854_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_855_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_856_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_857_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).

% bot_nat_0.extremum
tff(fact_858_Nat_Oadd__0__right,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),zero_zero_nat) = M2 ).

% Nat.add_0_right
tff(fact_859_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = zero_zero_nat )
    <=> ( ( M2 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_860_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% nat_add_left_cancel_less
tff(fact_861_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% nat_add_left_cancel_le
tff(fact_862_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_863_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_864_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_865_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_866_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_867_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_868_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_869_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_870_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_871_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_872_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_873_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_874_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_875_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_876_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_877_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_878_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_879_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: 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,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_880_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: 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,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_881_add__right__imp__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_882_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_883_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: 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),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_884_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: 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),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_885_add__left__imp__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_886_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_887_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: 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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.left_commute
tff(fact_888_add_Oleft__commute,axiom,
    ! [B: int,A: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.left_commute
tff(fact_889_add_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% add.left_commute
tff(fact_890_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C2)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% add.left_commute
tff(fact_891_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_892_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_893_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% add.commute
tff(fact_894_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_895_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_896_add_Oright__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_897_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_898_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_899_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_900_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_901_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.assoc
tff(fact_902_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.assoc
tff(fact_903_add_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% add.assoc
tff(fact_904_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% add.assoc
tff(fact_905_group__cancel_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_906_group__cancel_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_907_group__cancel_Oadd2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_908_group__cancel_Oadd2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K2),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B4) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_909_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_910_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_911_group__cancel_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_912_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(plus_plus_rat(A4),B) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_913_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_914_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_915_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_916_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_rat_rat(plus_plus_rat(I2),K2) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_917_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_918_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_919_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_920_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_921_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_922_infinite__descent,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ~ pp(aa_nat_bool(P2,N3))
         => ? [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N3))
              & ~ pp(aa_nat_bool(P2,M4)) ) )
     => pp(aa_nat_bool(P2,N)) ) ).

% infinite_descent
tff(fact_923_nat__less__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N3))
             => pp(aa_nat_bool(P2,M4)) )
         => pp(aa_nat_bool(P2,N3)) )
     => pp(aa_nat_bool(P2,N)) ) ).

% nat_less_induct
tff(fact_924_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_925_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_926_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( M2 != N ) ) ).

% less_not_refl2
tff(fact_927_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_928_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% nat_neq_iff
tff(fact_929_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun_nat_bool,K2: nat,B: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => ? [X4: nat] :
            ( pp(aa_nat_bool(P2,X4))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P2,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_930_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ).

% nat_le_linear
tff(fact_931_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( M2 = N ) ) ) ).

% le_antisym
tff(fact_932_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% eq_imp_le
tff(fact_933_le__trans,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),K2)) ) ) ).

% le_trans
tff(fact_934_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_935_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_936_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_937_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_938_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_939_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_940_add__le__imp__le__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_941_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_942_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_943_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_944_add__le__imp__le__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_945_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_946_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_947_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_948_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_949_add__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))) ) ).

% add_right_mono
tff(fact_950_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))) ) ).

% add_right_mono
tff(fact_951_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))) ) ).

% add_right_mono
tff(fact_952_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))) ) ).

% add_right_mono
tff(fact_953_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% less_eqE
tff(fact_954_add__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))) ) ).

% add_left_mono
tff(fact_955_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))) ) ).

% add_left_mono
tff(fact_956_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))) ) ).

% add_left_mono
tff(fact_957_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))) ) ).

% add_left_mono
tff(fact_958_add__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_mono
tff(fact_959_add__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_mono
tff(fact_960_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_mono
tff(fact_961_add__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_mono
tff(fact_962_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_963_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_964_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_965_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_966_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( ( I2 = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_967_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I2 = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_968_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_969_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( ( I2 = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_970_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),J))
        & ( K2 = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_971_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),J))
        & ( K2 = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_972_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
        & ( K2 = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_973_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
        & ( K2 = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_974_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_975_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_976_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_977_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_978_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_979_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_980_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_981_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_982_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_983_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_984_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_985_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_986_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_987_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_988_add__less__imp__less__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_989_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_990_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_991_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_992_add__less__imp__less__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_993_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_994_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_995_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_996_add__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))) ) ).

% add_strict_right_mono
tff(fact_997_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))) ) ).

% add_strict_right_mono
tff(fact_998_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))) ) ).

% add_strict_right_mono
tff(fact_999_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))) ) ).

% add_strict_right_mono
tff(fact_1000_add__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))) ) ).

% add_strict_left_mono
tff(fact_1001_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))) ) ).

% add_strict_left_mono
tff(fact_1002_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))) ) ).

% add_strict_left_mono
tff(fact_1003_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))) ) ).

% add_strict_left_mono
tff(fact_1004_add__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_strict_mono
tff(fact_1005_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_strict_mono
tff(fact_1006_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_strict_mono
tff(fact_1007_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_strict_mono
tff(fact_1008_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),J))
        & ( K2 = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1009_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),J))
        & ( K2 = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1010_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
        & ( K2 = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1011_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J))
        & ( K2 = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1012_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( ( I2 = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1013_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I2 = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1014_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1015_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( ( I2 = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1016_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1017_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1018_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1019_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1020_infinite__descent0,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( ~ pp(aa_nat_bool(P2,N3))
             => ? [M4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N3))
                  & ~ pp(aa_nat_bool(P2,M4)) ) ) )
       => pp(aa_nat_bool(P2,N)) ) ) ).

% infinite_descent0
tff(fact_1021_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1022_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_1023_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_1024_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1025_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_1026_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_1027_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1028_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1029_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1030_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_1031_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_1032_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( ( M2 != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% le_neq_implies_less
tff(fact_1033_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_or_eq_imp_le
tff(fact_1034_le__eq__less__or__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_1035_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_imp_le_nat
tff(fact_1036_nat__less__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        & ( M2 != N ) ) ) ).

% nat_less_le
tff(fact_1037_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = M2 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1038_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_1039_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% less_add_eq_less
tff(fact_1040_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),J))) ) ).

% trans_less_add2
tff(fact_1041_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M2))) ) ).

% trans_less_add1
tff(fact_1042_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ).

% add_less_mono1
tff(fact_1043_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),I2)) ).

% not_add_less2
tff(fact_1044_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),I2)) ).

% not_add_less1
tff(fact_1045_add__less__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_1046_add__lessD1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K2)) ) ).

% add_lessD1
tff(fact_1047_nat__le__iff__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K3) ) ).

% nat_le_iff_add
tff(fact_1048_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),J))) ) ).

% trans_le_add2
tff(fact_1049_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M2))) ) ).

% trans_le_add1
tff(fact_1050_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ).

% add_le_mono1
tff(fact_1051_add__le__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_le_mono
tff(fact_1052_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),L))
     => ? [N3: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N3) ) ).

% le_Suc_ex
tff(fact_1053_add__leD2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N)) ) ).

% add_leD2
tff(fact_1054_add__leD1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% add_leD1
tff(fact_1055_le__add2,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ).

% le_add2
tff(fact_1056_le__add1,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2))) ).

% le_add1
tff(fact_1057_add__leE,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N)) ) ) ).

% add_leE
tff(fact_1058_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)) ).

% max_add_distrib_right
tff(fact_1059_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(X),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_rat_rat(plus_plus_rat(X),Z)) ).

% max_add_distrib_right
tff(fact_1060_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),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,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% max_add_distrib_right
tff(fact_1061_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),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,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% max_add_distrib_right
tff(fact_1062_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_1063_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ).

% max_add_distrib_left
tff(fact_1064_max__add__distrib__left,axiom,
    ! [X: 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,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_1065_max__add__distrib__left,axiom,
    ! [X: 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,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_1066_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q2)) ).

% nat_add_max_right
tff(fact_1067_nat__add__max__left,axiom,
    ! [M2: nat,N: 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,M2),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).

% nat_add_max_left
tff(fact_1068_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1069_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1070_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1071_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1072_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1073_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1074_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1075_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1076_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_1077_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1078_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1079_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1080_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1081_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1082_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1083_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_1084_add__increasing2,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2))) ) ) ).

% add_increasing2
tff(fact_1085_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C2))) ) ) ).

% add_increasing2
tff(fact_1086_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2))) ) ) ).

% add_increasing2
tff(fact_1087_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2))) ) ) ).

% add_increasing2
tff(fact_1088_add__decreasing2,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1089_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1090_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1091_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1092_add__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2))) ) ) ).

% add_increasing
tff(fact_1093_add__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C2))) ) ) ).

% add_increasing
tff(fact_1094_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2))) ) ) ).

% add_increasing
tff(fact_1095_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2))) ) ) ).

% add_increasing
tff(fact_1096_add__decreasing,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1097_add__decreasing,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1098_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1099_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1100_add__less__le__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1101_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_less_le_mono
tff(fact_1102_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1103_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1104_add__le__less__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1105_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_le_less_mono
tff(fact_1106_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1107_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1108_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1109_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1110_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1111_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1112_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1113_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1114_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1115_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1116_pos__add__strict,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2))) ) ) ).

% pos_add_strict
tff(fact_1117_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2))) ) ) ).

% pos_add_strict
tff(fact_1118_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2))) ) ) ).

% pos_add_strict
tff(fact_1119_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2))) ) ) ).

% pos_add_strict
tff(fact_1120_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ ! [C: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1121_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_pos
tff(fact_1122_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_pos_pos
tff(fact_1123_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_pos
tff(fact_1124_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_pos
tff(fact_1125_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_1126_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_1127_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_1128_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_1129_ex__least__nat__le,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
       => ? [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
            & ! [I: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
               => ~ pp(aa_nat_bool(P2,I)) )
            & pp(aa_nat_bool(P2,K)) ) ) ) ).

% ex_least_nat_le
tff(fact_1130_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ? [K: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1131_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M2: nat,K2: nat] :
      ( ! [M: nat,N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M)),aa_nat_nat(F,N3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M2)),K2)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)))) ) ).

% mono_nat_linear_lb
tff(fact_1132_add__strict__increasing2,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1133_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1134_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1135_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1136_add__strict__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2))) ) ) ).

% add_strict_increasing
tff(fact_1137_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2))) ) ) ).

% add_strict_increasing
tff(fact_1138_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2))) ) ) ).

% add_strict_increasing
tff(fact_1139_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2))) ) ) ).

% add_strict_increasing
tff(fact_1140_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1141_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1142_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1143_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1144_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_1145_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_1146_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_1147_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_1148_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1149_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1150_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1151_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1152_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_1153_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_1154_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_1155_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_1156_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_1157_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_1158_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_1159_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_1160_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,set_Pr5648618587558075414at_nat(Xs))) ).

% List.finite_set
tff(fact_1161_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1162_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1163_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_1164_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_1165_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_1166_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_1167_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_1168_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(X),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_1169_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_1170_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1171_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_1172_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),M3)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1173_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M3: nat] :
        ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
          & pp(aa_set_nat_bool(member_nat(N2),S2)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1174_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),M3)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1175_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
    <=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(T),X))
        | pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).

% both_member_options_def
tff(fact_1176_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
       => ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
          | pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_1177_max__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ord_ma7524802468073614006at_nat(bot_bo2099793752762293965at_nat,X) = X ).

% max_bot
tff(fact_1178_max__bot,axiom,
    ! [X: set_real] : ord_max_set_real(bot_bot_set_real,X) = X ).

% max_bot
tff(fact_1179_max__bot,axiom,
    ! [X: set_nat] : ord_max_set_nat(bot_bot_set_nat,X) = X ).

% max_bot
tff(fact_1180_max__bot,axiom,
    ! [X: set_int] : ord_max_set_int(bot_bot_set_int,X) = X ).

% max_bot
tff(fact_1181_max__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X) = X ).

% max_bot
tff(fact_1182_max__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ord_ma7524802468073614006at_nat(X,bot_bo2099793752762293965at_nat) = X ).

% max_bot2
tff(fact_1183_max__bot2,axiom,
    ! [X: set_real] : ord_max_set_real(X,bot_bot_set_real) = X ).

% max_bot2
tff(fact_1184_max__bot2,axiom,
    ! [X: set_nat] : ord_max_set_nat(X,bot_bot_set_nat) = X ).

% max_bot2
tff(fact_1185_max__bot2,axiom,
    ! [X: set_int] : ord_max_set_int(X,bot_bot_set_int) = X ).

% max_bot2
tff(fact_1186_max__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),bot_bot_nat) = X ).

% max_bot2
tff(fact_1187_finite_OemptyI,axiom,
    pp(aa_set_complex_bool(finite3207457112153483333omplex,bot_bot_set_complex)) ).

% finite.emptyI
tff(fact_1188_finite_OemptyI,axiom,
    pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat)) ).

% finite.emptyI
tff(fact_1189_finite_OemptyI,axiom,
    pp(aa_set_real_bool(finite_finite_real,bot_bot_set_real)) ).

% finite.emptyI
tff(fact_1190_finite_OemptyI,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,bot_bot_set_nat)) ).

% finite.emptyI
tff(fact_1191_finite_OemptyI,axiom,
    pp(aa_set_int_bool(finite_finite_int,bot_bot_set_int)) ).

% finite.emptyI
tff(fact_1192_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1193_infinite__imp__nonempty,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
     => ( S2 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1194_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S2))
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_1195_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1196_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_1197_finite__transitivity__chain,axiom,
    ! [A4: set_set_nat,R: fun_se707272970763642173t_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( ! [X4: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),X4))
       => ( ! [X4: set_nat,Y3: set_nat,Z2: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y3))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,Y3),Z2))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Z2)) ) )
         => ( ! [X4: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(X4),A4))
               => ? [Y4: set_nat] :
                    ( pp(aa_set_set_nat_bool(member_set_nat(Y4),A4))
                    & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y4)) ) )
           => ( A4 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1198_finite__transitivity__chain,axiom,
    ! [A4: set_complex,R: fun_co1378794544173401837x_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] : ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),X4))
       => ( ! [X4: complex,Y3: complex,Z2: complex] :
              ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y3))
             => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y3),Z2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Z2)) ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),A4))
               => ? [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),A4))
                    & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y4)) ) )
           => ( A4 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1199_finite__transitivity__chain,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R: fun_Pr248683899324676269t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( ! [X4: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),X4))
       => ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),Y3))
             => ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,Y3),Z2))
               => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),Z2)) ) )
         => ( ! [X4: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A4))
               => ? [Y4: product_prod_nat_nat] :
                    ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y4),A4))
                    & pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X4),Y4)) ) )
           => ( A4 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1200_finite__transitivity__chain,axiom,
    ! [A4: set_real,R: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [X4: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),X4))
       => ( ! [X4: real,Y3: real,Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y3),Z2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Z2)) ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),A4))
               => ? [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),A4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y4)) ) )
           => ( A4 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1201_finite__transitivity__chain,axiom,
    ! [A4: set_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X4: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z2)) ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),A4))
               => ? [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),A4))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y4)) ) )
           => ( A4 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1202_finite__transitivity__chain,axiom,
    ! [A4: set_int,R: fun_int_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(R,X4),X4))
       => ( ! [X4: int,Y3: int,Z2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(R,Y3),Z2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Z2)) ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),A4))
               => ? [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),A4))
                    & pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y4)) ) )
           => ( A4 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1203_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A)) ).

% bot.extremum
tff(fact_1204_bot_Oextremum,axiom,
    ! [A: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A)) ).

% bot.extremum
tff(fact_1205_bot_Oextremum,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A)) ).

% bot.extremum
tff(fact_1206_bot_Oextremum,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A)) ).

% bot.extremum
tff(fact_1207_bot_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A)) ).

% bot.extremum
tff(fact_1208_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat))
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_1209_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1210_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1211_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1212_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1213_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat))
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1214_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1215_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1216_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1217_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1218_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A),bot_bo2099793752762293965at_nat)) ).

% bot.extremum_strict
tff(fact_1219_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A),bot_bot_set_real)) ).

% bot.extremum_strict
tff(fact_1220_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_1221_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_1222_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_1223_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,bot_bo2099793752762293965at_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1224_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(bot_bot_set_real),A)) ) ).

% bot.not_eq_extremum
tff(fact_1225_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,bot_bot_set_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1226_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A)) ) ).

% bot.not_eq_extremum
tff(fact_1227_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1228_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( A4 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1229_finite__has__maximal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A4)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1230_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A4))
     => ( ( A4 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A4))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A4))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1231_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A4))
     => ( ( A4 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A4))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A4))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1232_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1233_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( A4 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1234_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( A4 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1235_finite__has__minimal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A4)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1236_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A4))
     => ( ( A4 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A4))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A4))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1237_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A4))
     => ( ( A4 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A4))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A4))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1238_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1239_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( A4 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1240_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),X6))
           => ? [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),X6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Xa)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X6)) ) ) ).

% infinite_growing
tff(fact_1241_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),X6))
           => ? [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),X6))
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Xa)) ) )
       => ~ pp(aa_set_rat_bool(finite_finite_rat,X6)) ) ) ).

% infinite_growing
tff(fact_1242_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),X6))
           => ? [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),X6))
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Xa)) ) )
       => ~ pp(aa_set_num_bool(finite_finite_num,X6)) ) ) ).

% infinite_growing
tff(fact_1243_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),X6))
           => ? [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),X6))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ) ).

% infinite_growing
tff(fact_1244_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),X6))
           => ? [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),X6))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Xa)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X6)) ) ) ).

% infinite_growing
tff(fact_1245_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),S2))
            & ~ ? [Xa: real] :
                  ( pp(aa_set_real_bool(member_real(Xa),S2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1246_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,S2))
     => ( ( S2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),S2))
            & ~ ? [Xa: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa),S2))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1247_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,S2))
     => ( ( S2 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),S2))
            & ~ ? [Xa: num] :
                  ( pp(aa_set_num_bool(member_num(Xa),S2))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1248_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),S2))
            & ~ ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),S2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1249_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),S2))
            & ~ ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),S2))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1250_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1251_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X3)) ).

% linordered_field_no_lb
tff(fact_1252_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X3)) ).

% linordered_field_no_lb
tff(fact_1253_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X_12)) ).

% linordered_field_no_ub
tff(fact_1254_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),X_12)) ).

% linordered_field_no_ub
tff(fact_1255_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1256_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1257_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1258_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ).

% finite_subset
tff(fact_1259_finite__subset,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ).

% finite_subset
tff(fact_1260_finite__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
       => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4)) ) ) ).

% finite_subset
tff(fact_1261_finite__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ).

% finite_subset
tff(fact_1262_infinite__super,axiom,
    ! [S2: set_nat,T2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T2)) ) ) ).

% infinite_super
tff(fact_1263_infinite__super,axiom,
    ! [S2: set_complex,T2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T2)) ) ) ).

% infinite_super
tff(fact_1264_infinite__super,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,S2),T2))
     => ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2)) ) ) ).

% infinite_super
tff(fact_1265_infinite__super,axiom,
    ! [S2: set_int,T2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,T2)) ) ) ).

% infinite_super
tff(fact_1266_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ).

% rev_finite_subset
tff(fact_1267_rev__finite__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ).

% rev_finite_subset
tff(fact_1268_rev__finite__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4))
       => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4)) ) ) ).

% rev_finite_subset
tff(fact_1269_rev__finite__subset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
       => pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ).

% rev_finite_subset
tff(fact_1270_finite__psubset__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [A5: set_nat] :
            ( pp(aa_set_nat_bool(finite_finite_nat,A5))
           => ( ! [B5: set_nat] :
                  ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B5),A5))
                 => pp(aa_set_nat_bool(P2,B5)) )
             => pp(aa_set_nat_bool(P2,A5)) ) )
       => pp(aa_set_nat_bool(P2,A4)) ) ) ).

% finite_psubset_induct
tff(fact_1271_finite__psubset__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [A5: set_int] :
            ( pp(aa_set_int_bool(finite_finite_int,A5))
           => ( ! [B5: set_int] :
                  ( pp(aa_set_int_bool(ord_less_set_int(B5),A5))
                 => pp(aa_set_int_bool(P2,B5)) )
             => pp(aa_set_int_bool(P2,A5)) ) )
       => pp(aa_set_int_bool(P2,A4)) ) ) ).

% finite_psubset_induct
tff(fact_1272_finite__psubset__induct,axiom,
    ! [A4: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [A5: set_complex] :
            ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
           => ( ! [B5: set_complex] :
                  ( pp(aa_set_complex_bool(ord_less_set_complex(B5),A5))
                 => pp(aa_set_complex_bool(P2,B5)) )
             => pp(aa_set_complex_bool(P2,A5)) ) )
       => pp(aa_set_complex_bool(P2,A4)) ) ) ).

% finite_psubset_induct
tff(fact_1273_finite__psubset__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( ! [A5: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
           => ( ! [B5: set_Pr1261947904930325089at_nat] :
                  ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,B5),A5))
                 => pp(aa_set3135850010578026034t_bool(P2,B5)) )
             => pp(aa_set3135850010578026034t_bool(P2,A5)) ) )
       => pp(aa_set3135850010578026034t_bool(P2,A4)) ) ) ).

% finite_psubset_induct
tff(fact_1274_bounded__Max__nat,axiom,
    ! [P2: fun_nat_bool,X: nat,M5: nat] :
      ( pp(aa_nat_bool(P2,X))
     => ( ! [X4: nat] :
            ( pp(aa_nat_bool(P2,X4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M5)) )
       => ~ ! [M: nat] :
              ( pp(aa_nat_bool(P2,M))
             => ~ ! [X3: nat] :
                    ( pp(aa_nat_bool(P2,X3))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1275_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Xs)),B4))
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),set_Pr5648618587558075414at_nat(Xs)))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),B4)) ) ) ).

% subset_code(1)
tff(fact_1276_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B4))
    <=> ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(Xs)))
         => pp(aa_set_real_bool(member_real(X5),B4)) ) ) ).

% subset_code(1)
tff(fact_1277_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B4))
    <=> ! [X5: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X5),set_set_nat2(Xs)))
         => pp(aa_set_set_nat_bool(member_set_nat(X5),B4)) ) ) ).

% subset_code(1)
tff(fact_1278_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B4)
    <=> ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),B4)) ) ) ).

% subset_code(1)
tff(fact_1279_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),B4))
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X5),B4)) ) ) ).

% subset_code(1)
tff(fact_1280_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B4))
    <=> ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X5),B4)) ) ) ).

% subset_code(1)
tff(fact_1281_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : X != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1282_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va,Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1283_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1284_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A4))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),X4))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A4))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1285_finite__has__maximal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X4: set_int] :
            ( member_set_int(X4,A4)
            & pp(aa_set_int_bool(ord_less_eq_set_int(A),X4))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1286_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A4))
     => ( pp(aa_set_rat_bool(member_rat(A),A4))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A4))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X4))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A4))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1287_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A4))
     => ( pp(aa_set_num_bool(member_num(A),A4))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A4))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),X4))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A4))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1288_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1289_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1290_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),A))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1291_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A4))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),A))
            & ! [Xa: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa),A4))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1292_finite__has__minimal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X4: set_int] :
            ( member_set_int(X4,A4)
            & pp(aa_set_int_bool(ord_less_eq_set_int(X4),A))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A4)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1293_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A4))
     => ( pp(aa_set_rat_bool(member_rat(A),A4))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A4))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),A))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A4))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1294_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A4))
     => ( pp(aa_set_num_bool(member_num(A),A4))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A4))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),A))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A4))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1295_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),A))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1296_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),A))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1297_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A4)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A4 ) ).

% finite_list
tff(fact_1298_finite__list,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A4 ) ).

% finite_list
tff(fact_1299_finite__list,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ? [Xs2: list_int] : set_int2(Xs2) = A4 ) ).

% finite_list
tff(fact_1300_finite__list,axiom,
    ! [A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A4 ) ).

% finite_list
tff(fact_1301_finite__list,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ? [Xs2: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(Xs2) = A4 ) ).

% finite_list
tff(fact_1302_unbounded__k__infinite,axiom,
    ! [K2: nat,S2: set_nat] :
      ( ! [M: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),M))
         => ? [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N5))
              & pp(aa_set_nat_bool(member_nat(N5),S2)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S2)) ) ).

% unbounded_k_infinite
tff(fact_1303_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% bounded_nat_set_is_finite
tff(fact_1304_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M3: nat] :
        ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
          & pp(aa_set_nat_bool(member_nat(N2),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1305_empty__subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A4)) ).

% empty_subsetI
tff(fact_1306_empty__subsetI,axiom,
    ! [A4: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A4)) ).

% empty_subsetI
tff(fact_1307_empty__subsetI,axiom,
    ! [A4: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A4)) ).

% empty_subsetI
tff(fact_1308_empty__subsetI,axiom,
    ! [A4: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A4)) ).

% empty_subsetI
tff(fact_1309_subset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),bot_bo2099793752762293965at_nat))
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_1310_subset__empty,axiom,
    ! [A4: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),bot_bot_set_real))
    <=> ( A4 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1311_subset__empty,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),bot_bot_set_nat))
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1312_subset__empty,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),bot_bot_set_int))
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1313_empty__iff,axiom,
    ! [C2: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(C2),bot_bot_set_set_nat)) ).

% empty_iff
tff(fact_1314_empty__iff,axiom,
    ! [C2: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C2),bot_bo2099793752762293965at_nat)) ).

% empty_iff
tff(fact_1315_empty__iff,axiom,
    ! [C2: real] : ~ pp(aa_set_real_bool(member_real(C2),bot_bot_set_real)) ).

% empty_iff
tff(fact_1316_empty__iff,axiom,
    ! [C2: nat] : ~ pp(aa_set_nat_bool(member_nat(C2),bot_bot_set_nat)) ).

% empty_iff
tff(fact_1317_empty__iff,axiom,
    ! [C2: int] : ~ pp(aa_set_int_bool(member_int(C2),bot_bot_set_int)) ).

% empty_iff
tff(fact_1318_all__not__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ! [X5: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X5),A4))
    <=> ( A4 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_1319_all__not__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_1320_all__not__in__conv,axiom,
    ! [A4: set_real] :
      ( ! [X5: real] : ~ pp(aa_set_real_bool(member_real(X5),A4))
    <=> ( A4 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_1321_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ! [X5: nat] : ~ pp(aa_set_nat_bool(member_nat(X5),A4))
    <=> ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_1322_all__not__in__conv,axiom,
    ! [A4: set_int] :
      ( ! [X5: int] : ~ pp(aa_set_int_bool(member_int(X5),A4))
    <=> ( A4 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_1323_Collect__empty__eq,axiom,
    ! [P2: fun_list_nat_bool] :
      ( ( collect_list_nat(P2) = bot_bot_set_list_nat )
    <=> ! [X5: list_nat] : ~ pp(aa_list_nat_bool(P2,X5)) ) ).

% Collect_empty_eq
tff(fact_1324_Collect__empty__eq,axiom,
    ! [P2: fun_set_nat_bool] :
      ( ( collect_set_nat(P2) = bot_bot_set_set_nat )
    <=> ! [X5: set_nat] : ~ pp(aa_set_nat_bool(P2,X5)) ) ).

% Collect_empty_eq
tff(fact_1325_Collect__empty__eq,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool] :
      ( ( collec3392354462482085612at_nat(P2) = bot_bo2099793752762293965at_nat )
    <=> ! [X5: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(P2,X5)) ) ).

% Collect_empty_eq
tff(fact_1326_Collect__empty__eq,axiom,
    ! [P2: fun_real_bool] :
      ( ( collect_real(P2) = bot_bot_set_real )
    <=> ! [X5: real] : ~ pp(aa_real_bool(P2,X5)) ) ).

% Collect_empty_eq
tff(fact_1327_Collect__empty__eq,axiom,
    ! [P2: fun_nat_bool] :
      ( ( collect_nat(P2) = bot_bot_set_nat )
    <=> ! [X5: nat] : ~ pp(aa_nat_bool(P2,X5)) ) ).

% Collect_empty_eq
tff(fact_1328_Collect__empty__eq,axiom,
    ! [P2: fun_int_bool] :
      ( ( collect_int(P2) = bot_bot_set_int )
    <=> ! [X5: int] : ~ pp(aa_int_bool(P2,X5)) ) ).

% Collect_empty_eq
tff(fact_1329_empty__Collect__eq,axiom,
    ! [P2: fun_list_nat_bool] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P2) )
    <=> ! [X5: list_nat] : ~ pp(aa_list_nat_bool(P2,X5)) ) ).

% empty_Collect_eq
tff(fact_1330_empty__Collect__eq,axiom,
    ! [P2: fun_set_nat_bool] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P2) )
    <=> ! [X5: set_nat] : ~ pp(aa_set_nat_bool(P2,X5)) ) ).

% empty_Collect_eq
tff(fact_1331_empty__Collect__eq,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(P2) )
    <=> ! [X5: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(P2,X5)) ) ).

% empty_Collect_eq
tff(fact_1332_empty__Collect__eq,axiom,
    ! [P2: fun_real_bool] :
      ( ( bot_bot_set_real = collect_real(P2) )
    <=> ! [X5: real] : ~ pp(aa_real_bool(P2,X5)) ) ).

% empty_Collect_eq
tff(fact_1333_empty__Collect__eq,axiom,
    ! [P2: fun_nat_bool] :
      ( ( bot_bot_set_nat = collect_nat(P2) )
    <=> ! [X5: nat] : ~ pp(aa_nat_bool(P2,X5)) ) ).

% empty_Collect_eq
tff(fact_1334_empty__Collect__eq,axiom,
    ! [P2: fun_int_bool] :
      ( ( bot_bot_set_int = collect_int(P2) )
    <=> ! [X5: int] : ~ pp(aa_int_bool(P2,X5)) ) ).

% empty_Collect_eq
tff(fact_1335_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(F,lattic8794016678065449205x_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1336_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,lattic8440615504127631091l_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1337_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1338_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,lattic2675449441010098035t_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1339_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1340_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1341_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1342_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1343_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),S2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_complex_num(F,X3)),aa_complex_num(F,lattic1922116423962787043ex_num(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1344_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),S2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,lattic1613168225601753569al_num(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1345_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))),aa_complex_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1346_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S2))),aa_real_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1347_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))),aa_nat_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1348_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))),aa_int_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1349_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S2))),aa_complex_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1350_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S2))),aa_real_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1351_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_set_nat_nat(lattic4004264746738138117at_num(F),S2))),aa_nat_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1352_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,lattic5003618458639192673nt_num(F,S2))),aa_int_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1353_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( S2 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S2))),aa_complex_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1354_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,lattic5055836439445974935al_nat(F,S2))),aa_real_nat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1355_Euclid__induct,axiom,
    ! [P2: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,B3),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),zero_zero_nat))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_1356_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_1357_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_1358_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_1359_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_1360_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_1361_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1362_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1363_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1364_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_1365_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_1366_psubsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
     => ( ( A4 != B4 )
       => pp(aa_set_int_bool(ord_less_set_int(A4),B4)) ) ) ).

% psubsetI
tff(fact_1367_subset__iff__psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
        | ( A4 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1368_subset__psubset__trans,axiom,
    ! [A4: set_int,B4: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
     => ( pp(aa_set_int_bool(ord_less_set_int(B4),C4))
       => pp(aa_set_int_bool(ord_less_set_int(A4),C4)) ) ) ).

% subset_psubset_trans
tff(fact_1369_subset__not__subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4)) ) ) ).

% subset_not_subset_eq
tff(fact_1370_psubset__subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),C4))
       => pp(aa_set_int_bool(ord_less_set_int(A4),C4)) ) ) ).

% psubset_subset_trans
tff(fact_1371_psubset__imp__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4)) ) ).

% psubset_imp_subset
tff(fact_1372_psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
        & ( A4 != B4 ) ) ) ).

% psubset_eq
tff(fact_1373_psubsetE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
         => pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4)) ) ) ).

% psubsetE
tff(fact_1374_psubsetD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A4),B4))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C2),A4))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C2),B4)) ) ) ).

% psubsetD
tff(fact_1375_psubsetD,axiom,
    ! [A4: set_real,B4: set_real,C2: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A4),B4))
     => ( pp(aa_set_real_bool(member_real(C2),A4))
       => pp(aa_set_real_bool(member_real(C2),B4)) ) ) ).

% psubsetD
tff(fact_1376_psubsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A4),B4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C2),A4))
       => pp(aa_set_set_nat_bool(member_set_nat(C2),B4)) ) ) ).

% psubsetD
tff(fact_1377_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A4),B4))
     => ( pp(aa_set_nat_bool(member_nat(C2),A4))
       => pp(aa_set_nat_bool(member_nat(C2),B4)) ) ) ).

% psubsetD
tff(fact_1378_psubsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
     => ( pp(aa_set_int_bool(member_int(C2),A4))
       => pp(aa_set_int_bool(member_int(C2),B4)) ) ) ).

% psubsetD
tff(fact_1379_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo6469247547558333820t_bool) ).

% bot_set_def
tff(fact_1380_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo8343409700046576086t_bool) ).

% bot_set_def
tff(fact_1381_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo3782518310583830573t_bool) ).

% bot_set_def
tff(fact_1382_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bo4181092155253064880l_bool) ).

% bot_set_def
tff(fact_1383_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_bool) ).

% bot_set_def
tff(fact_1384_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_bool) ).

% bot_set_def
tff(fact_1385_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1386_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% verit_la_disequality
tff(fact_1387_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% verit_la_disequality
tff(fact_1388_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% verit_la_disequality
tff(fact_1389_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% verit_la_disequality
tff(fact_1390_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1391_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1392_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1393_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1394_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1395_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1396_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1397_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1398_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1399_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1400_ex__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ? [X5: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X5),A4))
    <=> ( A4 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1401_ex__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ? [X5: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
    <=> ( A4 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_1402_ex__in__conv,axiom,
    ! [A4: set_real] :
      ( ? [X5: real] : pp(aa_set_real_bool(member_real(X5),A4))
    <=> ( A4 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1403_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ? [X5: nat] : pp(aa_set_nat_bool(member_nat(X5),A4))
    <=> ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1404_ex__in__conv,axiom,
    ! [A4: set_int] :
      ( ? [X5: int] : pp(aa_set_int_bool(member_int(X5),A4))
    <=> ( A4 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1405_equals0I,axiom,
    ! [A4: set_set_nat] :
      ( ! [Y3: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(Y3),A4))
     => ( A4 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1406_equals0I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y3),A4))
     => ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_1407_equals0I,axiom,
    ! [A4: set_real] :
      ( ! [Y3: real] : ~ pp(aa_set_real_bool(member_real(Y3),A4))
     => ( A4 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1408_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] : ~ pp(aa_set_nat_bool(member_nat(Y3),A4))
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1409_equals0I,axiom,
    ! [A4: set_int] :
      ( ! [Y3: int] : ~ pp(aa_set_int_bool(member_int(Y3),A4))
     => ( A4 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1410_equals0D,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( A4 = bot_bot_set_set_nat )
     => ~ pp(aa_set_set_nat_bool(member_set_nat(A),A4)) ) ).

% equals0D
tff(fact_1411_equals0D,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A4)) ) ).

% equals0D
tff(fact_1412_equals0D,axiom,
    ! [A4: set_real,A: real] :
      ( ( A4 = bot_bot_set_real )
     => ~ pp(aa_set_real_bool(member_real(A),A4)) ) ).

% equals0D
tff(fact_1413_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ pp(aa_set_nat_bool(member_nat(A),A4)) ) ).

% equals0D
tff(fact_1414_equals0D,axiom,
    ! [A4: set_int,A: int] :
      ( ( A4 = bot_bot_set_int )
     => ~ pp(aa_set_int_bool(member_int(A),A4)) ) ).

% equals0D
tff(fact_1415_emptyE,axiom,
    ! [A: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(A),bot_bot_set_set_nat)) ).

% emptyE
tff(fact_1416_emptyE,axiom,
    ! [A: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat)) ).

% emptyE
tff(fact_1417_emptyE,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(member_real(A),bot_bot_set_real)) ).

% emptyE
tff(fact_1418_emptyE,axiom,
    ! [A: nat] : ~ pp(aa_set_nat_bool(member_nat(A),bot_bot_set_nat)) ).

% emptyE
tff(fact_1419_emptyE,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(member_int(A),bot_bot_set_int)) ).

% emptyE
tff(fact_1420_not__psubset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A4),bot_bo2099793752762293965at_nat)) ).

% not_psubset_empty
tff(fact_1421_not__psubset__empty,axiom,
    ! [A4: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A4),bot_bot_set_real)) ).

% not_psubset_empty
tff(fact_1422_not__psubset__empty,axiom,
    ! [A4: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A4),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_1423_not__psubset__empty,axiom,
    ! [A4: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A4),bot_bot_set_int)) ).

% not_psubset_empty
tff(fact_1424_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_1425_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_1426_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_1427_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_1428_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_1429_subset__emptyI,axiom,
    ! [A4: set_set_nat] :
      ( ! [X4: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X4),A4))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),bot_bot_set_set_nat)) ) ).

% subset_emptyI
tff(fact_1430_subset__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A4))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),bot_bo2099793752762293965at_nat)) ) ).

% subset_emptyI
tff(fact_1431_subset__emptyI,axiom,
    ! [A4: set_real] :
      ( ! [X4: real] : ~ pp(aa_set_real_bool(member_real(X4),A4))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A4),bot_bot_set_real)) ) ).

% subset_emptyI
tff(fact_1432_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X4: nat] : ~ pp(aa_set_nat_bool(member_nat(X4),A4))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),bot_bot_set_nat)) ) ).

% subset_emptyI
tff(fact_1433_subset__emptyI,axiom,
    ! [A4: set_int] :
      ( ! [X4: int] : ~ pp(aa_set_int_bool(member_int(X4),A4))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A4),bot_bot_set_int)) ) ).

% subset_emptyI
tff(fact_1434_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1435_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D2))
       => ? [E: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1436_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_1437_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_1438_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_1439_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_1440_complete__interval,axiom,
    ! [A: real,B: real,P2: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P2,A))
       => ( ~ pp(aa_real_bool(P2,B))
         => ? [C: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
              & ! [X3: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),C)) )
                 => pp(aa_real_bool(P2,X3)) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),D3)) )
                     => pp(aa_real_bool(P2,X4)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D3),C)) ) ) ) ) ) ).

% complete_interval
tff(fact_1441_complete__interval,axiom,
    ! [A: nat,B: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P2,A))
       => ( ~ pp(aa_nat_bool(P2,B))
         => ? [C: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
              & ! [X3: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),C)) )
                 => pp(aa_nat_bool(P2,X3)) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D3)) )
                     => pp(aa_nat_bool(P2,X4)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D3),C)) ) ) ) ) ) ).

% complete_interval
tff(fact_1442_complete__interval,axiom,
    ! [A: int,B: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P2,A))
       => ( ~ pp(aa_int_bool(P2,B))
         => ? [C: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
              & ! [X3: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),C)) )
                 => pp(aa_int_bool(P2,X3)) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),D3)) )
                     => pp(aa_int_bool(P2,X4)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D3),C)) ) ) ) ) ) ).

% complete_interval
tff(fact_1443_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),T)) ) ).

% pinf(6)
tff(fact_1444_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),T)) ) ).

% pinf(6)
tff(fact_1445_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),T)) ) ).

% pinf(6)
tff(fact_1446_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),T)) ) ).

% pinf(6)
tff(fact_1447_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T)) ) ).

% pinf(6)
tff(fact_1448_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X3)) ) ).

% pinf(8)
tff(fact_1449_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X3)) ) ).

% pinf(8)
tff(fact_1450_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X3)) ) ).

% pinf(8)
tff(fact_1451_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X3)) ) ).

% pinf(8)
tff(fact_1452_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3)) ) ).

% pinf(8)
tff(fact_1453_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),T)) ) ).

% minf(6)
tff(fact_1454_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),T)) ) ).

% minf(6)
tff(fact_1455_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),T)) ) ).

% minf(6)
tff(fact_1456_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),T)) ) ).

% minf(6)
tff(fact_1457_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T)) ) ).

% minf(6)
tff(fact_1458_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X3)) ) ).

% minf(8)
tff(fact_1459_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X3)) ) ).

% minf(8)
tff(fact_1460_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X3)) ) ).

% minf(8)
tff(fact_1461_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X3)) ) ).

% minf(8)
tff(fact_1462_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3)) ) ).

% minf(8)
tff(fact_1463_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_1464_le__numeral__extra_I3_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).

% le_numeral_extra(3)
tff(fact_1465_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_1466_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_1467_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2,S3) ) ).

% deg_SUcn_Node
tff(fact_1468_count__notin,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( aa_Pro4243173770044397973at_nat(count_4203492906077236349at_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1469_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => ( aa_real_nat(count_list_real(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1470_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1471_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1472_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => ( aa_int_nat(count_list_int(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1473_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => ( aa_nat_nat(count_list_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1474_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3)
     => ~ ! [N3: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) ) ).

% even_odd_cases
tff(fact_1475_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_1476_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_1477_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_1478_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_1479_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_less_eq
tff(fact_1480_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ).

% Suc_le_mono
tff(fact_1481_add__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% add_Suc_right
tff(fact_1482_max__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),N)) ).

% max_Suc_Suc
tff(fact_1483_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1484_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_1485_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_1486_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_1487_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_1488_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] : X != aa_nat_nat(suc,N3) ) ).

% list_decode.cases
tff(fact_1489_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_1490_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1491_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1492_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1493_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1494_nat__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,N3))
           => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) )
       => pp(aa_nat_bool(P2,N)) ) ) ).

% nat_induct
tff(fact_1495_diff__induct,axiom,
    ! [P2: fun_nat_fun_nat_bool,M2: nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,X4),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M2),N)) ) ) ) ).

% diff_induct
tff(fact_1496_zero__induct,axiom,
    ! [P2: fun_nat_bool,K2: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P2,N3)) )
       => pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_1497_Suc__neq__Zero,axiom,
    ! [M2: nat] : aa_nat_nat(suc,M2) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1498_Zero__neq__Suc,axiom,
    ! [M2: nat] : zero_zero_nat != aa_nat_nat(suc,M2) ).

% Zero_neq_Suc
tff(fact_1499_Zero__not__Suc,axiom,
    ! [M2: nat] : zero_zero_nat != aa_nat_nat(suc,M2) ).

% Zero_not_Suc
tff(fact_1500_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M: nat] : N = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_1501_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K2))
     => ( ( K2 != aa_nat_nat(suc,I2) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
             => ( K2 != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1502_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_lessD
tff(fact_1503_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I2)),K2))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
           => ( K2 != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1504_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( ( aa_nat_nat(suc,M2) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),N)) ) ) ).

% Suc_lessI
tff(fact_1505_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( M2 = N ) ) ) ).

% less_SucE
tff(fact_1506_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_1507_Ex__less__Suc,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P2,I4)) )
    <=> ( pp(aa_nat_bool(P2,N))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P2,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_1508_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) ) ) ).

% less_Suc_eq
tff(fact_1509_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2))) ) ).

% not_less_eq
tff(fact_1510_All__less__Suc,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P2,I4)) )
    <=> ( pp(aa_nat_bool(P2,N))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P2,I4)) ) ) ) ).

% All_less_Suc
tff(fact_1511_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
    <=> ? [M6: nat] :
          ( ( M2 = aa_nat_nat(suc,M6) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M6)) ) ) ).

% Suc_less_eq2
tff(fact_1512_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
       => ( M2 = N ) ) ) ).

% less_antisym
tff(fact_1513_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_less_SucD
tff(fact_1514_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I2)),K2)) ) ) ).

% less_trans_Suc
tff(fact_1515_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P2: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( ! [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I3),aa_nat_nat(suc,I3)))
       => ( ! [I3: nat,J2: nat,K: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I3),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,J2),K))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I3),K)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I2),J)) ) ) ) ).

% less_Suc_induct
tff(fact_1516_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => pp(aa_nat_bool(P2,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J))
             => ( pp(aa_nat_bool(P2,aa_nat_nat(suc,I3)))
               => pp(aa_nat_bool(P2,I3)) ) )
         => pp(aa_nat_bool(P2,I2)) ) ) ) ).

% strict_inc_induct
tff(fact_1517_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
      <=> ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1518_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% Suc_leD
tff(fact_1519_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( M2 = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_1520_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_1521_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M7))
     => ? [M: nat] : M7 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_1522_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        | ( M2 = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_1523_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_1524_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M2)) ) ).

% not_less_eq_eq
tff(fact_1525_full__nat__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M4)),N3))
             => pp(aa_nat_bool(P2,M4)) )
         => pp(aa_nat_bool(P2,N3)) )
     => pp(aa_nat_bool(P2,N)) ) ).

% full_nat_induct
tff(fact_1526_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(P2,M2))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N3))
             => ( pp(aa_nat_bool(P2,N3))
               => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_1527_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z2)) ) )
         => ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N3),aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M2),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1528_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(suc,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_1529_add__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% add_Suc
tff(fact_1530_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_1531_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1532_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1533_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N3)),aa_nat_num(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1534_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1535_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N3)),aa_nat_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_1536_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1537_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1538_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N3)),aa_nat_num(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1539_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1540_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N3)),aa_nat_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1541_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N3)),aa_nat_set_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1542_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1543_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1544_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1545_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N6))) ) ) ).

% lift_Suc_mono_le
tff(fact_1546_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N3))),aa_nat_set_int(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N6)),aa_nat_set_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1547_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N3))),aa_nat_rat(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N6)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1548_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N3))),aa_nat_num(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N6)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1549_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N3))),aa_nat_nat(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N6)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1550_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N3))),aa_nat_int(F,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N6)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_1551_Ex__less__Suc2,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P2,I4)) )
    <=> ( pp(aa_nat_bool(P2,zero_zero_nat))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P2,aa_nat_nat(suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1552_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_1553_All__less__Suc2,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P2,I4)) )
    <=> ( pp(aa_nat_bool(P2,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P2,aa_nat_nat(suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_1554_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M: nat] : N = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_1555_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> ( ( M2 = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M2 = aa_nat_nat(suc,J3) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1556_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)) ) ).

% Suc_leI
tff(fact_1557_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_le_eq
tff(fact_1558_dec__induct,axiom,
    ! [I2: nat,J: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(P2,I2))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J))
               => ( pp(aa_nat_bool(P2,N3))
                 => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) ) )
         => pp(aa_nat_bool(P2,J)) ) ) ) ).

% dec_induct
tff(fact_1559_inc__induct,axiom,
    ! [I2: nat,J: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(P2,J))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J))
               => ( pp(aa_nat_bool(P2,aa_nat_nat(suc,N3)))
                 => pp(aa_nat_bool(P2,N3)) ) ) )
         => pp(aa_nat_bool(P2,I2)) ) ) ) ).

% inc_induct
tff(fact_1560_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_le_lessD
tff(fact_1561_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
      <=> ( N = M2 ) ) ) ).

% le_less_Suc_eq
tff(fact_1562_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_Suc_eq_le
tff(fact_1563_less__eq__Suc__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M2)) ) ).

% less_eq_Suc_le
tff(fact_1564_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_1565_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1566_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) )
    <=> ( ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1567_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q3)) ) ).

% less_natE
tff(fact_1568_less__add__Suc1,axiom,
    ! [I2: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2)))) ).

% less_add_Suc1
tff(fact_1569_less__add__Suc2,axiom,
    ! [I2: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),I2)))) ).

% less_add_Suc2
tff(fact_1570_less__iff__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K3)) ) ).

% less_iff_Suc_add
tff(fact_1571_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ? [K: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K)) ) ).

% less_imp_Suc_add
tff(fact_1572_ex__least__nat__less,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
       => ? [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),N))
            & ! [I: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K))
               => ~ pp(aa_nat_bool(P2,I)) )
            & pp(aa_nat_bool(P2,aa_nat_nat(suc,K))) ) ) ) ).

% ex_least_nat_less
tff(fact_1573_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X3)) ) ).

% minf(7)
tff(fact_1574_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X3)) ) ).

% minf(7)
tff(fact_1575_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X3)) ) ).

% minf(7)
tff(fact_1576_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X3)) ) ).

% minf(7)
tff(fact_1577_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3)) ) ).

% minf(7)
tff(fact_1578_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),T)) ) ).

% minf(5)
tff(fact_1579_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),T)) ) ).

% minf(5)
tff(fact_1580_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),T)) ) ).

% minf(5)
tff(fact_1581_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),T)) ) ).

% minf(5)
tff(fact_1582_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T)) ) ).

% minf(5)
tff(fact_1583_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1584_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1585_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1586_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1587_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1588_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1589_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1590_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1591_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1592_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1593_minf_I2_J,axiom,
    ! [P2: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P2,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
           => ( ( pp(aa_real_bool(P2,X3))
                | pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P3,X3))
                | pp(aa_real_bool(Q4,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1594_minf_I2_J,axiom,
    ! [P2: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q4: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
         => ( pp(aa_rat_bool(P2,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q4,X4)) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
           => ( ( pp(aa_rat_bool(P2,X3))
                | pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P3,X3))
                | pp(aa_rat_bool(Q4,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1595_minf_I2_J,axiom,
    ! [P2: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q4: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
         => ( pp(aa_num_bool(P2,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q4,X4)) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
           => ( ( pp(aa_num_bool(P2,X3))
                | pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P3,X3))
                | pp(aa_num_bool(Q4,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1596_minf_I2_J,axiom,
    ! [P2: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P2,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
           => ( ( pp(aa_nat_bool(P2,X3))
                | pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P3,X3))
                | pp(aa_nat_bool(Q4,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1597_minf_I2_J,axiom,
    ! [P2: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P2,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
           => ( ( pp(aa_int_bool(P2,X3))
                | pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P3,X3))
                | pp(aa_int_bool(Q4,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1598_minf_I1_J,axiom,
    ! [P2: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P2,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
           => ( ( pp(aa_real_bool(P2,X3))
                & pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P3,X3))
                & pp(aa_real_bool(Q4,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1599_minf_I1_J,axiom,
    ! [P2: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q4: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
         => ( pp(aa_rat_bool(P2,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q4,X4)) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
           => ( ( pp(aa_rat_bool(P2,X3))
                & pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P3,X3))
                & pp(aa_rat_bool(Q4,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1600_minf_I1_J,axiom,
    ! [P2: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q4: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
         => ( pp(aa_num_bool(P2,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q4,X4)) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
           => ( ( pp(aa_num_bool(P2,X3))
                & pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P3,X3))
                & pp(aa_num_bool(Q4,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1601_minf_I1_J,axiom,
    ! [P2: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P2,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
           => ( ( pp(aa_nat_bool(P2,X3))
                & pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P3,X3))
                & pp(aa_nat_bool(Q4,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1602_minf_I1_J,axiom,
    ! [P2: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P2,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
           => ( ( pp(aa_int_bool(P2,X3))
                & pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P3,X3))
                & pp(aa_int_bool(Q4,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1603_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X3)) ) ).

% pinf(7)
tff(fact_1604_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X3)) ) ).

% pinf(7)
tff(fact_1605_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X3)) ) ).

% pinf(7)
tff(fact_1606_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X3)) ) ).

% pinf(7)
tff(fact_1607_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3)) ) ).

% pinf(7)
tff(fact_1608_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),T)) ) ).

% pinf(5)
tff(fact_1609_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),T)) ) ).

% pinf(5)
tff(fact_1610_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),T)) ) ).

% pinf(5)
tff(fact_1611_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),T)) ) ).

% pinf(5)
tff(fact_1612_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T)) ) ).

% pinf(5)
tff(fact_1613_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1614_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1615_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1616_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1617_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1618_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1619_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1620_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1621_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1622_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1623_pinf_I2_J,axiom,
    ! [P2: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P2,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
           => ( ( pp(aa_real_bool(P2,X3))
                | pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P3,X3))
                | pp(aa_real_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1624_pinf_I2_J,axiom,
    ! [P2: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q4: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
         => ( pp(aa_rat_bool(P2,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q4,X4)) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
           => ( ( pp(aa_rat_bool(P2,X3))
                | pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P3,X3))
                | pp(aa_rat_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1625_pinf_I2_J,axiom,
    ! [P2: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q4: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
         => ( pp(aa_num_bool(P2,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q4,X4)) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
           => ( ( pp(aa_num_bool(P2,X3))
                | pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P3,X3))
                | pp(aa_num_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1626_pinf_I2_J,axiom,
    ! [P2: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P2,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
           => ( ( pp(aa_nat_bool(P2,X3))
                | pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P3,X3))
                | pp(aa_nat_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1627_pinf_I2_J,axiom,
    ! [P2: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P2,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
           => ( ( pp(aa_int_bool(P2,X3))
                | pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P3,X3))
                | pp(aa_int_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1628_pinf_I1_J,axiom,
    ! [P2: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q4: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P2,X4))
          <=> pp(aa_real_bool(P3,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q4,X4)) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
           => ( ( pp(aa_real_bool(P2,X3))
                & pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P3,X3))
                & pp(aa_real_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1629_pinf_I1_J,axiom,
    ! [P2: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q4: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
         => ( pp(aa_rat_bool(P2,X4))
          <=> pp(aa_rat_bool(P3,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q4,X4)) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
           => ( ( pp(aa_rat_bool(P2,X3))
                & pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P3,X3))
                & pp(aa_rat_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1630_pinf_I1_J,axiom,
    ! [P2: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q4: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
         => ( pp(aa_num_bool(P2,X4))
          <=> pp(aa_num_bool(P3,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q4,X4)) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
           => ( ( pp(aa_num_bool(P2,X3))
                & pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P3,X3))
                & pp(aa_num_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1631_pinf_I1_J,axiom,
    ! [P2: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q4: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P2,X4))
          <=> pp(aa_nat_bool(P3,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q4,X4)) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
           => ( ( pp(aa_nat_bool(P2,X3))
                & pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P3,X3))
                & pp(aa_nat_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1632_pinf_I1_J,axiom,
    ! [P2: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q4: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P2,X4))
          <=> pp(aa_int_bool(P3,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q4,X4)) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
           => ( ( pp(aa_int_bool(P2,X3))
                & pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P3,X3))
                & pp(aa_int_bool(Q4,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1633_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B3))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A)) ) ).

% ex_gt_or_lt
tff(fact_1634_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% is_num_normalize(1)
tff(fact_1635_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% is_num_normalize(1)
tff(fact_1636_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% is_num_normalize(1)
tff(fact_1637_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(produc6161850002892822231at_nat(R2,S),R)
     => ( ( S4 = S )
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1638_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( member1466754251312161552at_nat(produc9060074326276436823at_nat(R2,S),R)
     => ( ( S4 = S )
       => member1466754251312161552at_nat(produc9060074326276436823at_nat(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1639_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(R2,S)),R))
     => ( ( S4 = S )
       => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(R2,S4)),R)) ) ) ).

% ssubst_Pair_rhs
tff(fact_1640_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S)),R))
     => ( ( S4 = S )
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S4)),R)) ) ) ).

% ssubst_Pair_rhs
tff(fact_1641_ssubst__Pair__rhs,axiom,
    ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S4: int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(R2,S),R)
     => ( ( S4 = S )
       => member5262025264175285858nt_int(product_Pair_int_int(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1642_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),X) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_1643_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_1644_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1645_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X2: num] : size_option_num(X,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1646_bot__empty__eq,axiom,
    ! [X3: set_nat] :
      ( pp(aa_set_nat_bool(bot_bo8343409700046576086t_bool,X3))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X3),bot_bot_set_set_nat)) ) ).

% bot_empty_eq
tff(fact_1647_bot__empty__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(bot_bo3782518310583830573t_bool,X3))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),bot_bo2099793752762293965at_nat)) ) ).

% bot_empty_eq
tff(fact_1648_bot__empty__eq,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(bot_bo4181092155253064880l_bool,X3))
    <=> pp(aa_set_real_bool(member_real(X3),bot_bot_set_real)) ) ).

% bot_empty_eq
tff(fact_1649_bot__empty__eq,axiom,
    ! [X3: nat] :
      ( pp(aa_nat_bool(bot_bot_fun_nat_bool,X3))
    <=> pp(aa_set_nat_bool(member_nat(X3),bot_bot_set_nat)) ) ).

% bot_empty_eq
tff(fact_1650_bot__empty__eq,axiom,
    ! [X3: int] :
      ( pp(aa_int_bool(bot_bot_fun_int_bool,X3))
    <=> pp(aa_set_int_bool(member_int(X3),bot_bot_set_int)) ) ).

% bot_empty_eq
tff(fact_1651_Collect__empty__eq__bot,axiom,
    ! [P2: fun_list_nat_bool] :
      ( ( collect_list_nat(P2) = bot_bot_set_list_nat )
    <=> ( P2 = bot_bo6469247547558333820t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1652_Collect__empty__eq__bot,axiom,
    ! [P2: fun_set_nat_bool] :
      ( ( collect_set_nat(P2) = bot_bot_set_set_nat )
    <=> ( P2 = bot_bo8343409700046576086t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1653_Collect__empty__eq__bot,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool] :
      ( ( collec3392354462482085612at_nat(P2) = bot_bo2099793752762293965at_nat )
    <=> ( P2 = bot_bo3782518310583830573t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1654_Collect__empty__eq__bot,axiom,
    ! [P2: fun_real_bool] :
      ( ( collect_real(P2) = bot_bot_set_real )
    <=> ( P2 = bot_bo4181092155253064880l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1655_Collect__empty__eq__bot,axiom,
    ! [P2: fun_nat_bool] :
      ( ( collect_nat(P2) = bot_bot_set_nat )
    <=> ( P2 = bot_bot_fun_nat_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1656_Collect__empty__eq__bot,axiom,
    ! [P2: fun_int_bool] :
      ( ( collect_int(P2) = bot_bot_set_int )
    <=> ( P2 = bot_bot_fun_int_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1657_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1658_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1659_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_1660_exists__least__lemma,axiom,
    ! [P2: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ? [X_1: nat] : pp(aa_nat_bool(P2,X_1))
       => ? [N3: nat] :
            ( ~ pp(aa_nat_bool(P2,N3))
            & pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) ) ) ).

% exists_least_lemma
tff(fact_1661__092_060open_062vebt__insert_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Ix_M_Amax_Ami_Ama_J_J_Adeg_A_ItreeList_091high_Ami_An_A_058_061_Avebt__insert_A_ItreeList_A_B_Ahigh_Ami_An_J_A_Ilow_Ami_An_J_093_J_A_Iif_AminNull_A_ItreeList_A_B_Ahigh_Ami_An_J_Athen_Avebt__insert_Asummary_A_Ihigh_Ami_An_J_Aelse_Asummary_J_092_060close_062,axiom,
    vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,mi),ma))),deg,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))),vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),summary)) ).

% \<open>vebt_insert (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (x, max mi ma)) deg (treeList[high mi n := vebt_insert (treeList ! high mi n) (low mi n)]) (if minNull (treeList ! high mi n) then vebt_insert summary (high mi n) else summary)\<close>
tff(fact_1662_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_1663_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_1664_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ).

% pair_less_iff1
tff(fact_1665_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_1666_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),I2,Y) = list_u1324408373059187874T_VEBT(Xs,I2,Y) ).

% list_update_overwrite
tff(fact_1667_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y5) ) ).

% not_None_eq
tff(fact_1668_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y5: num] : X = some_num(Y5) ) ).

% not_None_eq
tff(fact_1669_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y5)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_1670_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y5: num] : X != some_num(Y5)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_1671_list__update__id,axiom,
    ! [Xs: list_nat,I2: nat] : list_update_nat(Xs,I2,aa_nat_nat(nth_nat(Xs),I2)) = Xs ).

% list_update_id
tff(fact_1672_list__update__id,axiom,
    ! [Xs: list_int,I2: nat] : list_update_int(Xs,I2,aa_nat_int(nth_int(Xs),I2)) = Xs ).

% list_update_id
tff(fact_1673_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat] : list_u1324408373059187874T_VEBT(Xs,I2,nth_VEBT_VEBT(Xs,I2)) = Xs ).

% list_update_id
tff(fact_1674_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I2 != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_1675_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_int,X: int] :
      ( ( I2 != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I2,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_1676_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I2 != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_1677_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1678_list__update__swap,axiom,
    ! [I2: nat,I5: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X7: vEBT_VEBT] :
      ( ( I2 != I5 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),I5,X7) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I5,X7),I2,X) ) ) ).

% list_update_swap
tff(fact_1679_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1680_size__neq__size__imp__neq,axiom,
    ! [X: list_bool,Y: list_bool] :
      ( ( size_size_list_bool(X) != size_size_list_bool(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1681_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X) != size_size_list_nat(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1682_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1683_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1684_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_1685_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_1686_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_1687_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_1688_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_1689_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != some_num(X22) ) ).

% option.exhaust
tff(fact_1690_split__option__ex,axiom,
    ! [P2: fun_op2627489687811228755t_bool] :
      ( ? [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P2,X_13))
    <=> ( pp(aa_opt6800557945374683724t_bool(P2,none_P5556105721700978146at_nat))
        | ? [X5: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P2,some_P7363390416028606310at_nat(X5))) ) ) ).

% split_option_ex
tff(fact_1691_split__option__ex,axiom,
    ! [P2: fun_option_num_bool] :
      ( ? [X_13: option_num] : pp(aa_option_num_bool(P2,X_13))
    <=> ( pp(aa_option_num_bool(P2,none_num))
        | ? [X5: num] : pp(aa_option_num_bool(P2,some_num(X5))) ) ) ).

% split_option_ex
tff(fact_1692_split__option__all,axiom,
    ! [P2: fun_op2627489687811228755t_bool] :
      ( ! [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P2,X_13))
    <=> ( pp(aa_opt6800557945374683724t_bool(P2,none_P5556105721700978146at_nat))
        & ! [X5: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P2,some_P7363390416028606310at_nat(X5))) ) ) ).

% split_option_all
tff(fact_1693_split__option__all,axiom,
    ! [P2: fun_option_num_bool] :
      ( ! [X_13: option_num] : pp(aa_option_num_bool(P2,X_13))
    <=> ( pp(aa_option_num_bool(P2,none_num))
        & ! [X5: num] : pp(aa_option_num_bool(P2,some_num(X5))) ) ) ).

% split_option_all
tff(fact_1694_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: fun_op5682630191948829869t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1695_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: fun_op775672374849087042m_bool,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1696_combine__options__cases,axiom,
    ! [X: option_num,P2: fun_op7732454363416259258t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1697_combine__options__cases,axiom,
    ! [X: option_num,P2: fun_op1962577433978643573m_bool,Y: option_num] :
      ( ( ( X = none_num )
       => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) )
       => ( ! [A3: num,B3: num] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1698_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1699_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1700_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1701_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1702_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I2: nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Xs)),A4))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A4))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I2,X))),A4)) ) ) ).

% set_update_subsetI
tff(fact_1703_set__update__subsetI,axiom,
    ! [Xs: list_real,A4: set_real,X: real,I2: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I2,X))),A4)) ) ) ).

% set_update_subsetI
tff(fact_1704_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A4: set_set_nat,X: set_nat,I2: nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),A4))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I2,X))),A4)) ) ) ).

% set_update_subsetI
tff(fact_1705_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,X: nat,I2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),A4))
     => ( pp(aa_set_nat_bool(member_nat(X),A4))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I2,X))),A4)) ) ) ).

% set_update_subsetI
tff(fact_1706_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,X: vEBT_VEBT,I2: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A4)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A4))
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I2,X)),A4) ) ) ).

% set_update_subsetI
tff(fact_1707_set__update__subsetI,axiom,
    ! [Xs: list_int,A4: set_int,X: int,I2: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I2,X))),A4)) ) ) ).

% set_update_subsetI
tff(fact_1708_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_1709_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1710_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X4,Y3),R2)
         => member8206827879206165904at_nat(produc6161850002892822231at_nat(X4,Y3),S) )
     => ord_le3000389064537975527at_nat(R2,S) ) ).

% subrelI
tff(fact_1711_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( member1466754251312161552at_nat(produc9060074326276436823at_nat(X4,Y3),R2)
         => member1466754251312161552at_nat(produc9060074326276436823at_nat(X4,Y3),S) )
     => ord_le5997549366648089703at_nat(R2,S) ) ).

% subrelI
tff(fact_1712_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X4,Y3)),R2))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X4,Y3)),S)) )
     => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,R2),S)) ) ).

% subrelI
tff(fact_1713_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)),R2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)),S)) )
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,R2),S)) ) ).

% subrelI
tff(fact_1714_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y3: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X4,Y3),R2)
         => member5262025264175285858nt_int(product_Pair_int_int(X4,Y3),S) )
     => ord_le2843351958646193337nt_int(R2,S) ) ).

% subrelI
tff(fact_1715_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1716_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_1717_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).

% in_children_def
tff(fact_1718_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P2: fun_Pr7757163693017731481t_bool,N: nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),set_Pr5648618587558075414at_nat(Xs)))
         => pp(aa_Pro2232559814647311250t_bool(P2,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
       => pp(aa_Pro2232559814647311250t_bool(P2,nth_Pr7617993195940197384at_nat(Xs,N))) ) ) ).

% inthall
tff(fact_1719_inthall,axiom,
    ! [Xs: list_real,P2: fun_real_bool,N: nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(P2,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P2,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_1720_inthall,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_bool,N: nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),set_set_nat2(Xs)))
         => pp(aa_set_nat_bool(P2,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P2,nth_set_nat(Xs,N))) ) ) ).

% inthall
tff(fact_1721_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_1722_inthall,axiom,
    ! [Xs: list_bool,P2: fun_bool_bool,N: nat] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_bool_bool(P2,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_1723_inthall,axiom,
    ! [Xs: list_nat,P2: fun_nat_bool,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(P2,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_1724_inthall,axiom,
    ! [Xs: list_int,P2: fun_int_bool,N: nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_int_bool(P2,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_1725_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_1726_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_1727__C4_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [X3: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),na) = I )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),X3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),ma)) ) ) ) ) ) ).

% "4.hyps"(9)
tff(fact_1728_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] : X != 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] : X != 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] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_1729_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S3),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3),X4)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_1730_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),X4)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),X4)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1731_intind,axiom,
    ! [I2: nat,N: nat,P2: fun_nat_bool,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( pp(aa_nat_bool(P2,X))
       => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(replicate_nat(N,X)),I2))) ) ) ).

% intind
tff(fact_1732_intind,axiom,
    ! [I2: nat,N: nat,P2: fun_int_bool,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( pp(aa_int_bool(P2,X))
       => pp(aa_int_bool(P2,aa_nat_int(nth_int(replicate_int(N,X)),I2))) ) ) ).

% intind
tff(fact_1733_intind,axiom,
    ! [I2: nat,N: nat,P2: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( pp(aa_VEBT_VEBT_bool(P2,X))
       => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I2))) ) ) ).

% intind
tff(fact_1734_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_1735_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_1736__C4_Ohyps_C_I2_J,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "4.hyps"(2)
tff(fact_1737__C4_Ohyps_C_I8_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).

% "4.hyps"(8)
tff(fact_1738__C4_Oprems_C_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).

% "4.prems"(1)
tff(fact_1739__C4_Oprems_C_I2_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).

% "4.prems"(2)
tff(fact_1740_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_1741__C4_OIH_C_I1_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X3),na))
        & ! [Xa: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
           => ! [Xb: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
               => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(X3,Xa)),Xb))
                 => ( pp(aa_nat_bool(vEBT_vebt_member(X3),Xb))
                    | ( Xa = Xb ) ) ) ) ) ) ) ).

% "4.IH"(1)
tff(fact_1742__C4_Ohyps_C_I5_J,axiom,
    ! [I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I)),X_13))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I)) ) ) ).

% "4.hyps"(5)
tff(fact_1743_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))) ) ).

% high_bound_aux
tff(fact_1744_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% member_bound
tff(fact_1745_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_1746_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_1747_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_1748_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_1749_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_1750_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_1751_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_1752_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_1753_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_1754_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_1755_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M2),N)) ).

% numeral_plus_numeral
tff(fact_1756_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M2),N)) ).

% numeral_plus_numeral
tff(fact_1757_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M2),N)) ).

% numeral_plus_numeral
tff(fact_1758_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(M2),N)) ).

% numeral_plus_numeral
tff(fact_1759_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M2),N)) ).

% numeral_plus_numeral
tff(fact_1760_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W2)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1761_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W2)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1762_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W2)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1763_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: extended_enat] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2),Z)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(aa_num_num(plus_plus_num(V),W2)),Z) ).

% add_numeral_left
tff(fact_1764_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W2))),Z) ).

% add_numeral_left
tff(fact_1765_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_1766_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1767_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1768_replicate__eq__replicate,axiom,
    ! [M2: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M2,X) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M2 = N )
        & ( ( M2 != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1769_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X)) = N ).

% length_replicate
tff(fact_1770_length__replicate,axiom,
    ! [N: nat,X: bool] : size_size_list_bool(replicate_bool(N,X)) = N ).

% length_replicate
tff(fact_1771_length__replicate,axiom,
    ! [N: nat,X: nat] : size_size_list_nat(replicate_nat(N,X)) = N ).

% length_replicate
tff(fact_1772_length__replicate,axiom,
    ! [N: nat,X: int] : size_size_list_int(replicate_int(N,X)) = N ).

% length_replicate
tff(fact_1773_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_1774_length__list__update,axiom,
    ! [Xs: list_bool,I2: nat,X: bool] : size_size_list_bool(list_update_bool(Xs,I2,X)) = size_size_list_bool(Xs) ).

% length_list_update
tff(fact_1775_length__list__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat] : size_size_list_nat(list_update_nat(Xs,I2,X)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_1776_length__list__update,axiom,
    ! [Xs: list_int,I2: nat,X: int] : size_size_list_int(list_update_int(Xs,I2,X)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_1777_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_1778__C4_OIH_C_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
       => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(summary,X)),Y))
         => ( pp(aa_nat_bool(vEBT_vebt_member(summary),Y))
            | ( X = Y ) ) ) ) ) ).

% "4.IH"(2)
tff(fact_1779_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_1780_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(1)
tff(fact_1781_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(1)
tff(fact_1782_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( 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)) = aa_num_nat(numeral_numeral_nat,V) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( 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)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).

% max_number_of(1)
tff(fact_1783_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(1)
tff(fact_1784_max__0__1_I4_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(4)
tff(fact_1785_max__0__1_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(4)
tff(fact_1786_max__0__1_I4_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X)),zero_zero_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(4)
tff(fact_1787_max__0__1_I4_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_1788_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),zero_zero_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(4)
tff(fact_1789_max__0__1_I3_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(3)
tff(fact_1790_max__0__1_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(3)
tff(fact_1791_max__0__1_I3_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(3)
tff(fact_1792_max__0__1_I3_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_1793_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(3)
tff(fact_1794_xyprop,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))) ) ).

% xyprop
tff(fact_1795__092_060open_062low_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ay_An_A_060_A2_A_094_An_092_060close_062,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ya),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))) ) ).

% \<open>low x n < 2 ^ n \<and> low y n < 2 ^ n\<close>
tff(fact_1796_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P2: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(replicate_int(N,A))))
         => pp(aa_int_bool(P2,X5)) )
    <=> ( pp(aa_int_bool(P2,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1797_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P2: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P2,X5)) )
    <=> ( pp(aa_nat_bool(P2,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1798_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
         => pp(aa_VEBT_VEBT_bool(P2,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P2,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1799_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P2: fun_int_bool] :
      ( ? [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(replicate_int(N,A))))
          & pp(aa_int_bool(P2,X5)) )
    <=> ( pp(aa_int_bool(P2,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1800_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P2: fun_nat_bool] :
      ( ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P2,X5)) )
    <=> ( pp(aa_nat_bool(P2,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1801_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( ? [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
          & pp(aa_VEBT_VEBT_bool(P2,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P2,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1802_in__set__replicate,axiom,
    ! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1803_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(replicate_real(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1804_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(replicate_set_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1805_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(replicate_int(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1806_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1807_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1808_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I2))
     => ( list_u1324408373059187874T_VEBT(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_1809_list__update__beyond,axiom,
    ! [Xs: list_bool,I2: nat,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I2))
     => ( list_update_bool(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_1810_list__update__beyond,axiom,
    ! [Xs: list_nat,I2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I2))
     => ( list_update_nat(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_1811_list__update__beyond,axiom,
    ! [Xs: list_int,I2: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),I2))
     => ( list_update_int(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_1812_nth__replicate,axiom,
    ! [I2: nat,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( aa_nat_nat(nth_nat(replicate_nat(N,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_1813_nth__replicate,axiom,
    ! [I2: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( aa_nat_int(nth_int(replicate_int(N,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_1814_nth__replicate,axiom,
    ! [I2: nat,N: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I2) = X ) ) ).

% nth_replicate
tff(fact_1815__C00_C,axiom,
    ( ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),na))
    & ( na = m )
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))
    & ( size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na) ) ) ).

% "00"
tff(fact_1816_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_1817_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I2,X)),I2))
      <=> pp(X) ) ) ).

% nth_list_update_eq
tff(fact_1818_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_1819_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_1820_mimaxyprop,axiom,
    ( ~ ( ( xa = mi )
        | ( xa = ma ) )
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)))
    & ( size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na) ) ) ).

% mimaxyprop
tff(fact_1821_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_1822_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_1823_set__swap,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I2))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_1824_set__swap,axiom,
    ! [I2: nat,Xs: list_bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(list_update_bool(Xs,I2,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I2))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_1825_set__swap,axiom,
    ! [I2: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I2,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I2))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_1826_set__swap,axiom,
    ! [I2: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I2,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I2))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_1827_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = N ).

% Ex_list_of_length
tff(fact_1828_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).

% Ex_list_of_length
tff(fact_1829_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_1830_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_1831_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_1832_neq__if__length__neq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) != size_size_list_bool(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1833_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_1834_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_1835_replicate__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( ( size_s5460976970255530739at_nat(Xs) = N )
     => ( ! [Y3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y3),set_Pr5648618587558075414at_nat(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replic4235873036481779905at_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1836_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y3: real] :
            ( pp(aa_set_real_bool(member_real(Y3),set_real2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_real(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1837_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N )
     => ( ! [Y3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(Y3),set_set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_set_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1838_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),set_VEBT_VEBT2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_VEBT_VEBT(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1839_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X: bool] :
      ( ( size_size_list_bool(Xs) = N )
     => ( ! [Y3: bool] :
            ( member_bool(Y3,set_bool2(Xs))
           => ( pp(Y3)
            <=> pp(X) ) )
       => ( Xs = replicate_bool(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1840_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( size_size_list_nat(Xs) = N )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1841_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),set_int2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_int(N,X) ) ) ) ).

% replicate_eqI
tff(fact_1842_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => ( X4 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1843_replicate__length__same,axiom,
    ! [Xs: list_bool,X: bool] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => ( pp(X4)
          <=> pp(X) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1844_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( X4 = X ) )
     => ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1845_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => ( X4 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1846_less__by__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4)) ) ).

% less_by_empty
tff(fact_1847_numeral__Bit0,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_Bit0
tff(fact_1848_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_1849_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_1850_numeral__Bit0,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit0,N)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)) ).

% numeral_Bit0
tff(fact_1851_numeral__Bit0,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_Bit0
tff(fact_1852_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_1853_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_1854_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1855_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1856_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1857_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1858_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1859_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_1860_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),N) ).

% Suc_nat_number_of_add
tff(fact_1861_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,D4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),D4)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_1862_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: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_1863_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_1864_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_1865_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_1866_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_1867_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_1868_zero__neq__numeral,axiom,
    ! [N: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(N) ).

% zero_neq_numeral
tff(fact_1869_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_1870_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M5)
     => ? [N3: nat] :
        ! [X3: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X3,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X3)),N3)) ) ) ).

% finite_maxlen
tff(fact_1871_finite__maxlen,axiom,
    ! [M5: set_list_bool] :
      ( finite7866827932528546246t_bool(M5)
     => ? [N3: nat] :
        ! [X3: list_bool] :
          ( member_list_bool(X3,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X3)),N3)) ) ) ).

% finite_maxlen
tff(fact_1872_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( finite8100373058378681591st_nat(M5)
     => ? [N3: nat] :
        ! [X3: list_nat] :
          ( member_list_nat(X3,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X3)),N3)) ) ) ).

% finite_maxlen
tff(fact_1873_finite__maxlen,axiom,
    ! [M5: set_list_int] :
      ( finite3922522038869484883st_int(M5)
     => ? [N3: nat] :
        ! [X3: list_int] :
          ( member_list_int(X3,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X3)),N3)) ) ) ).

% finite_maxlen
tff(fact_1874_length__induct,axiom,
    ! [P2: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P2,Ys2)) )
         => pp(aa_lis2822252094023462681T_bool(P2,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P2,Xs)) ) ).

% length_induct
tff(fact_1875_length__induct,axiom,
    ! [P2: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys2: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys2)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P2,Ys2)) )
         => pp(aa_list_bool_bool(P2,Xs2)) )
     => pp(aa_list_bool_bool(P2,Xs)) ) ).

% length_induct
tff(fact_1876_length__induct,axiom,
    ! [P2: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2)))
             => pp(aa_list_nat_bool(P2,Ys2)) )
         => pp(aa_list_nat_bool(P2,Xs2)) )
     => pp(aa_list_nat_bool(P2,Xs)) ) ).

% length_induct
tff(fact_1877_length__induct,axiom,
    ! [P2: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2)))
             => pp(aa_list_int_bool(P2,Ys2)) )
         => pp(aa_list_int_bool(P2,Xs2)) )
     => pp(aa_list_int_bool(P2,Xs)) ) ).

% length_induct
tff(fact_1878_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1879_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1880_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1881_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1882_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1883_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1884_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_1885_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_le_numeral
tff(fact_1886_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_le_numeral
tff(fact_1887_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_le_numeral
tff(fact_1888_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_le_numeral
tff(fact_1889_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_le_numeral
tff(fact_1890_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_le_zero
tff(fact_1891_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_1892_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_1893_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_1894_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_1895_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_less_numeral
tff(fact_1896_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_less_numeral
tff(fact_1897_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_less_numeral
tff(fact_1898_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_less_numeral
tff(fact_1899_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_less_numeral
tff(fact_1900_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_1901_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_1902_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_1903_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N)),zero_z5237406670263579293d_enat)) ).

% not_numeral_less_zero
tff(fact_1904_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_1905_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] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1906_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I4))
            <=> pp(aa_nat_bool(nth_bool(Ys),I4)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1907_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] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1908_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] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1909_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_na2351039247863542899T_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
         => ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P2,I4),X_13)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K2 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P2,I4),nth_VEBT_VEBT(Xs3,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_1910_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_na9137082442855928016l_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
         => ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P2,I4),X_13)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K2 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P2,I4),aa_nat_bool(nth_bool(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_1911_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_nat_fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I4),X_13)) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K2 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I4),aa_nat_nat(nth_nat(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_1912_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_nat_fun_int_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
         => ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P2,I4),X_13)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K2 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P2,I4),aa_nat_int(nth_int(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_1913_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1914_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I3))
            <=> pp(aa_nat_bool(nth_bool(Ys),I3)) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1915_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1916_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1917_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( ( aa_set_nat_nat(nat_set_encode,A4) = aa_set_nat_nat(nat_set_encode,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_1918_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_1919_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ! [I3: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_1920_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
     => ( ( ? [A3: bool,B3: bool] : A1 = vEBT_Leaf(A3,B3)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                     => ( ( M = N3 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                         => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                       => ( ( M = aa_nat_nat(suc,N3) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                           => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                         => ( ( M = N3 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                             => ( ! [I: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),X_13))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),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) = I )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                           => ( ( M = aa_nat_nat(suc,N3) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                               => ( ! [I: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                                     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),X_13))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),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) = I )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1921_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
    <=> ( ( ? [A6: bool,B6: bool] : A1 = vEBT_Leaf(A6,B6)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N2))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N2)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N2)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N2))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N2) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N2) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N2))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N2)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N2)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) )
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N2))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N2))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N2) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N2))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N2) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N2))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1922_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X)),size_s6755466524823107622T_VEBT(Xs))) ).

% count_le_length
tff(fact_1923_count__le__length,axiom,
    ! [Xs: list_bool,X: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(count_list_bool(Xs),X)),size_size_list_bool(Xs))) ).

% count_le_length
tff(fact_1924_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),X)),size_size_list_nat(Xs))) ).

% count_le_length
tff(fact_1925_count__le__length,axiom,
    ! [Xs: list_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),X)),size_size_list_int(Xs))) ).

% count_le_length
tff(fact_1926_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
         => ( ( M2 = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ! [I3: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_1927_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1928_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_1929_length__pos__if__in__set,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s5460976970255530739at_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_1930_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).

% length_pos_if_in_set
tff(fact_1931_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_1932_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_pos_if_in_set
tff(fact_1933_length__pos__if__in__set,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_bool(Xs))) ) ).

% length_pos_if_in_set
tff(fact_1934_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_1935_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).

% length_pos_if_in_set
tff(fact_1936_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_1937_nth__mem,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,N)),set_Pr5648618587558075414at_nat(Xs))) ) ).

% nth_mem
tff(fact_1938_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs))) ) ).

% nth_mem
tff(fact_1939_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(nth_set_nat(Xs,N)),set_set_nat2(Xs))) ) ).

% nth_mem
tff(fact_1940_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_1941_nth__mem,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(aa_nat_bool(nth_bool(Xs),N),set_bool2(Xs)) ) ).

% nth_mem
tff(fact_1942_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_1943_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs))) ) ).

% nth_mem
tff(fact_1944_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P2,X4)) )
       => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_1945_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P2: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( ! [X4: bool] :
            ( member_bool(X4,set_bool2(Xs))
           => pp(aa_bool_bool(P2,X4)) )
       => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_1946_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
           => pp(aa_nat_bool(P2,X4)) )
       => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_1947_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P2: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
           => pp(aa_int_bool(P2,X4)) )
       => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_1948_in__set__conv__nth,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
          & ( nth_Pr7617993195940197384at_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1949_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1950_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3254054031482475050et_nat(Xs)))
          & ( nth_set_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1951_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1952_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I4))
          <=> pp(X) ) ) ) ).

% in_set_conv_nth
tff(fact_1953_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1954_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_1955_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P2: fun_Pr7757163693017731481t_bool,X: product_prod_nat_nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs)))
         => pp(aa_Pro2232559814647311250t_bool(P2,nth_Pr7617993195940197384at_nat(Xs,I3))) )
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => pp(aa_Pro2232559814647311250t_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1956_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P2: fun_real_bool,X: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
         => pp(aa_real_bool(P2,aa_nat_real(nth_real(Xs),I3))) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => pp(aa_real_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1957_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_bool,X: set_nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P2,nth_set_nat(Xs,I3))) )
     => ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
       => pp(aa_set_nat_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1958_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,I3))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1959_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P2: fun_bool_bool,X: bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),I3))) )
     => ( member_bool(X,set_bool2(Xs))
       => pp(aa_bool_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1960_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: fun_nat_bool,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),I3))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1961_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P2: fun_int_bool,X: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
         => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),I3))) )
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => pp(aa_int_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_1962_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_1963_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P2: fun_bool_bool] :
      ( ! [X5: bool] :
          ( member_bool(X5,set_bool2(Xs))
         => pp(aa_bool_bool(P2,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_1964_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_nat_bool(P2,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_1965_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P2: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
         => pp(aa_int_bool(P2,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
         => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_1966_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_1967_set__update__memI,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_1968_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_1969_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(list_update_set_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_1970_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_1971_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(X,set_bool2(list_update_bool(Xs,N,X))) ) ).

% set_update_memI
tff(fact_1972_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_1973_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_1974_nth__list__update,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I2 = J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),J) = X ) )
        & ( ( I2 != J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_1975_nth__list__update,axiom,
    ! [I2: nat,Xs: list_bool,X: bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I2,X)),J))
      <=> ( ( ( I2 = J )
           => pp(X) )
          & ( ( I2 != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_1976_nth__list__update,axiom,
    ! [I2: nat,Xs: list_nat,J: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
     => ( ( ( I2 = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,X)),J) = X ) )
        & ( ( I2 != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_1977_nth__list__update,axiom,
    ! [I2: nat,Xs: list_int,J: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( ( ( I2 = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I2,X)),J) = X ) )
        & ( ( I2 != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I2,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_1978_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I2,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_1979_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I2,X) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I2))
        <=> pp(X) ) ) ) ).

% list_update_same_conv
tff(fact_1980_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I2,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_1981_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( ( list_update_int(Xs,I2,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_1982_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1983_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1984_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_nat(nat_set_encode,A4) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_1985_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1986_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1987_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1988_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1989_zero__less__power2,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1990_zero__less__power2,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_1991_zero__less__power2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1992_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1993_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1994_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1995_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1996_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1997_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_1998_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_1999_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_2000_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_2001_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_2002_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_2003_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_2004_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_2005_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_2006_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2007_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2008_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2009_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2010_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2011_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2012_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2013_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2014_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2015_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
            & pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% member_inv
tff(fact_2016_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_2017_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% high_def
tff(fact_2018_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2019_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2020_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_2021_divide__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2022_divide__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C2) = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2023_divide__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2024_divide__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C2),A) = aa_complex_complex(divide1717551699836669952omplex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2025_divide__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C2),A) = aa_rat_rat(divide_divide_rat(C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2026_divide__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2027_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_2028_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_2029_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_2030_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_2031_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_2032_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_2033_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2034_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_2035_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_2036_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_2037_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_2038_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_2039_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_2040_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2041_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2042_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_2043_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2044_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_2045_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_2046_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_2047_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_real ).

% power_zero_numeral
tff(fact_2048_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_2049_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_int ).

% power_zero_numeral
tff(fact_2050_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_2051_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2052_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_2053_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_2054_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M2 = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2055_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2056_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2057_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_2058_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)) ).

% add_divide_distrib
tff(fact_2059_add__divide__distrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) ).

% add_divide_distrib
tff(fact_2060_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2))) ) ) ).

% divide_right_mono_neg
tff(fact_2061_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_rat_rat(divide_divide_rat(A),C2))) ) ) ).

% divide_right_mono_neg
tff(fact_2062_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2063_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2064_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2065_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2066_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2067_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2068_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2069_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2070_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2071_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2072_divide__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))) ) ) ).

% divide_right_mono
tff(fact_2073_divide__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2))) ) ) ).

% divide_right_mono
tff(fact_2074_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2075_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2076_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2077_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2078_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2))) ) ) ).

% divide_strict_right_mono
tff(fact_2079_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))) ) ) ).

% divide_strict_right_mono
tff(fact_2080_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2081_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_2082_divide__less__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2083_divide__less__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2084_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_2085_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_2086_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2087_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2088_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_2089_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_2090_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_2091_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_2092_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2093_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2094_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_2095_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_2096_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2097_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2098_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2099_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2100_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_2101_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_2102_divide__le__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_2103_divide__le__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_2104_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_2105_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_2106_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_2107_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_2108_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_2109_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_2110_field__sum__of__halves,axiom,
    ! [X: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_2111_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_2112_half__gt__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2113_half__gt__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2114_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% half_gt_zero_iff
tff(fact_2115_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% half_gt_zero_iff
tff(fact_2116_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_2117_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_2118_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2119_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2120_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2121_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2122_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2123_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_le_power
tff(fact_2124_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_power
tff(fact_2125_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_le_power
tff(fact_2126_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_power
tff(fact_2127_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono
tff(fact_2128_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono
tff(fact_2129_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).

% power_mono
tff(fact_2130_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono
tff(fact_2131_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_less_power
tff(fact_2132_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_less_power
tff(fact_2133_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_less_power
tff(fact_2134_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_less_power
tff(fact_2135_nat__power__less__imp__less,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_2136_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2137_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2138_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2139_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2140_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2141_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2142_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2143_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2144_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2145_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2146_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2147_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2148_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_2149_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2150_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_2151_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2152_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_2153_power__gt__expt,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K2))) ) ).

% power_gt_expt
tff(fact_2154_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N))) ) ).

% nat_one_le_power
tff(fact_2155_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2156_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2157_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2158_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2159_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2160_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2161_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2162_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2163_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_2164_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_2165_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_2166_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_2167_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_2168_less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% less_exp
tff(fact_2169_power2__nat__le__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% power2_nat_le_imp_le
tff(fact_2170_power2__nat__le__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% power2_nat_le_eq_le
tff(fact_2171_self__le__ge2__pow,axiom,
    ! [K2: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M2))) ) ).

% self_le_ge2_pow
tff(fact_2172_zero__le__power2,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2173_zero__le__power2,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2174_zero__le__power2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_2175_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2176_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2177_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2178_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2179_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2180_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2181_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2182_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_2183_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2184_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2185_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2186_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2187_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_2188_power2__less__0,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)) ).

% power2_less_0
tff(fact_2189_power2__less__0,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_2190_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2191_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2192_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2193_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_2194_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2195_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2196_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2197_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2198_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2199_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2200_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2201_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2202_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2203_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_2204_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_2205_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_2206_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_2207_add__self__div__2,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),M2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M2 ).

% add_self_div_2
tff(fact_2208_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_2209_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit0,N))) ).

% semiring_norm(76)
tff(fact_2210_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_2211_div__by__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = M2 ).

% div_by_Suc_0
tff(fact_2212_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M2: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_2213_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2214_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_2215_div__exp__eq,axiom,
    ! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ).

% div_exp_eq
tff(fact_2216_div__exp__eq,axiom,
    ! [A: int,M2: nat,N: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ).

% div_exp_eq
tff(fact_2217_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).

% deg1Leaf
tff(fact_2218_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ).

% deg_1_Leaf
tff(fact_2219_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ) ).

% deg_1_Leafy
tff(fact_2220_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_2221_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_2222_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_2223_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_2224_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_2225_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_2226_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_2227_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_2228_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_2229_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_2230_semiring__norm_I78_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M2)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% semiring_norm(78)
tff(fact_2231_semiring__norm_I75_J,axiom,
    ! [M2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),one)) ).

% semiring_norm(75)
tff(fact_2232_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_2233_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2234_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2235_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2236_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2237_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_2238_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_2239_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_2240_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_2241_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2242_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2243_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2244_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2245_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2246_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_2247_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_2248_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_2249_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2250_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2251_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2252_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2253_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2254_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2255_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2256_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2257_power__inject__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2258_power__inject__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( ( aa_nat_rat(power_power_rat(A),M2) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2259_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2260_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( aa_nat_int(power_power_int(A),M2) = aa_nat_int(power_power_int(A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2261_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_2262_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_2263_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_2264_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_2265_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_2266_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_2267_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_2268_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_2269_max__0__1_I6_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(6)
tff(fact_2270_max__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(6)
tff(fact_2271_max__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X)),one_one_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(6)
tff(fact_2272_max__0__1_I6_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(6)
tff(fact_2273_max__0__1_I6_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),one_one_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(6)
tff(fact_2274_max__0__1_I5_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(5)
tff(fact_2275_max__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(5)
tff(fact_2276_max__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(5)
tff(fact_2277_max__0__1_I5_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(5)
tff(fact_2278_max__0__1_I5_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(5)
tff(fact_2279_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_2280_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_2281_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% divide_le_0_1_iff
tff(fact_2282_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2283_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2284_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% divide_less_0_1_iff
tff(fact_2285_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_2286_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2287_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2288_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2289_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2290_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2291_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2292_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2293_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2294_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2295_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2296_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2297_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2298_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2299_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2300_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2301_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2302_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2303_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2304_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2305_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2306_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2307_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2308_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2309_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2310_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2311_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2312_one__add__one,axiom,
    plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2313_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2314_power__strict__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2315_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M2)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2316_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2317_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2318_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2319_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2320_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2321_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2322_one__plus__numeral,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_2323_one__plus__numeral,axiom,
    ! [N: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_2324_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_2325_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_2326_one__plus__numeral,axiom,
    ! [N: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_2327_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_2328_numeral__plus__one,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_2329_numeral__plus__one,axiom,
    ! [N: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_2330_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_2331_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_2332_numeral__plus__one,axiom,
    ! [N: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_2333_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_2334_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2335_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),one_one_real))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2336_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2337_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2338_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2339_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2340_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2341_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2342_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2343_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2344_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_2345_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2346_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_2347_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2348_power__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2349_power__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M2)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2350_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2351_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2352_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_2353_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_2354_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_2355_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_2356_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_2357_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_2358_le__numeral__extra_I4_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).

% le_numeral_extra(4)
tff(fact_2359_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_2360_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_2361_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2362_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2363_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2364_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2365_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2366_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_2367_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_2368_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_2369_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_2370_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_2371_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_2372_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_2373_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_2374_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_2375_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_2376_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_2377_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_2378_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2379_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2380_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2381_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2382_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_2383_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_2384_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_2385_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_2386_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_2387_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_2388_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_2389_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_2390_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_2391_zero__less__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_2392_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_2393_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_2394_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_2395_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_2396_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_2397_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_2398_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N))) ).

% one_le_numeral
tff(fact_2399_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,N))) ).

% one_le_numeral
tff(fact_2400_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% one_le_numeral
tff(fact_2401_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% one_le_numeral
tff(fact_2402_one__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).

% one_le_numeral
tff(fact_2403_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_2404_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_2405_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).

% not_numeral_less_one
tff(fact_2406_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat)) ).

% not_numeral_less_one
tff(fact_2407_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_2408_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real))) ).

% less_add_one
tff(fact_2409_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat))) ).

% less_add_one
tff(fact_2410_less__add__one,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).

% less_add_one
tff(fact_2411_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).

% less_add_one
tff(fact_2412_add__mono1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real))) ) ).

% add_mono1
tff(fact_2413_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat))) ) ).

% add_mono1
tff(fact_2414_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2415_add__mono1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_2416_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_2417_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2418_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2419_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_2420_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_2421_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_2422_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_2423_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_2424_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2425_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% one_le_power
tff(fact_2426_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% one_le_power
tff(fact_2427_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% one_le_power
tff(fact_2428_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).

% one_le_power
tff(fact_2429_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),M2)) ) ) ).

% div_less_dividend
tff(fact_2430_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = M2 )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2431_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2432_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_2433_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2434_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2435_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2436_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2437_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_2438_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2439_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_2440_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_2441_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_2442_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_2443_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2444_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(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_2445_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2446_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_2447_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_2448_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_2449_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2450_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2451_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_2452_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_2453_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B)) ) ).

% gt_half_sum
tff(fact_2454_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B)) ) ).

% gt_half_sum
tff(fact_2455_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)))) ) ).

% less_half_sum
tff(fact_2456_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)))) ) ).

% less_half_sum
tff(fact_2457_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_2458_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_2459_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_2460_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_2461_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_2462_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2463_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2464_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2465_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2466_power__less__imp__less__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2467_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2468_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2469_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2470_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2471_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2472_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2473_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2474_power__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4))) ) ) ).

% power_increasing
tff(fact_2475_power__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).

% power_increasing
tff(fact_2476_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).

% power_increasing
tff(fact_2477_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).

% power_increasing
tff(fact_2478_nat__induct__non__zero,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P2,one_one_nat))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(P2,N3))
               => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_2479_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_2480_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: bool,B: bool] :
      ( ( ( X = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(fTrue,B) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,fTrue) ) )
          & ( ( X != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,B) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_2481_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2482_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2483_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2484_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_2485_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_2486_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2487_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2488_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2489_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2490_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_2491_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_2492_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_2493_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_2494_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2495_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2496_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2497_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2498_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_decreasing
tff(fact_2499_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2500_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N))) ) ) ) ).

% power_decreasing
tff(fact_2501_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2502_power__le__imp__le__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2503_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2504_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2505_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2506_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% self_le_power
tff(fact_2507_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% self_le_power
tff(fact_2508_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% self_le_power
tff(fact_2509_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).

% self_le_power
tff(fact_2510_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% one_less_power
tff(fact_2511_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% one_less_power
tff(fact_2512_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% one_less_power
tff(fact_2513_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).

% one_less_power
tff(fact_2514_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_2515_div__le__mono,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K2))) ) ).

% div_le_mono
tff(fact_2516_div__le__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),M2)) ).

% div_le_dividend
tff(fact_2517_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N3)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_2518_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K2))
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N3)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_2519_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2520_Suc__div__le__mono,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M2)),N))) ).

% Suc_div_le_mono
tff(fact_2521_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),M2))) ) ) ).

% div_le_mono2
tff(fact_2522_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_2523_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2524_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2525_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2526_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2527_nat__induct2,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( pp(aa_nat_bool(P2,one_one_nat))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P2,N3))
             => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_induct2
tff(fact_2528_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_2529_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_2530_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_2531_enat__ord__number_I1_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_2532_enat__ord__number_I2_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M2)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_2533_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% pos2
tff(fact_2534_discrete,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).

% discrete
tff(fact_2535_discrete,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).

% discrete
tff(fact_2536_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2537_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_2538_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) ) ) ).

% div_positive
tff(fact_2539_div__positive,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))) ) ) ).

% div_positive
tff(fact_2540_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% zle_add1_eq_le
tff(fact_2541_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N))
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_2542_mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(times_times_complex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2543_mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2544_mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C2) = aa_rat_rat(times_times_rat(B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2545_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2546_mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2547_mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A) = aa_complex_complex(times_times_complex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2548_mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2549_mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A) = aa_rat_rat(times_times_rat(C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2550_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2551_mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2552_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_2553_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2554_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_2555_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_2556_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_2557_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2558_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2559_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2560_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_2561_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_2562_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_2563_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_2564_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_2565_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_2566_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_2567_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2568_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_2569_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_2570_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_2571_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_2572_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_2573_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_2574_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_2575_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_2576_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_2577_times__divide__eq__left,axiom,
    ! [B: complex,C2: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C2)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C2) ).

% times_divide_eq_left
tff(fact_2578_times__divide__eq__left,axiom,
    ! [B: rat,C2: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C2)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C2) ).

% times_divide_eq_left
tff(fact_2579_times__divide__eq__left,axiom,
    ! [B: real,C2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C2) ).

% times_divide_eq_left
tff(fact_2580_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C2) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C2)) ).

% divide_divide_eq_left
tff(fact_2581_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C2) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C2)) ).

% divide_divide_eq_left
tff(fact_2582_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% divide_divide_eq_left
tff(fact_2583_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),B) ).

% divide_divide_eq_right
tff(fact_2584_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C2)),B) ).

% divide_divide_eq_right
tff(fact_2585_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B) ).

% divide_divide_eq_right
tff(fact_2586_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C2) ).

% times_divide_eq_right
tff(fact_2587_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C2) ).

% times_divide_eq_right
tff(fact_2588_times__divide__eq__right,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) ).

% times_divide_eq_right
tff(fact_2589_mult__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2) )
    <=> ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2590_mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N) )
    <=> ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2591_mult__0__right,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2592_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = zero_zero_nat )
    <=> ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2593_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
    <=> ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2594_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = one_one_nat )
    <=> ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2595_mult__cancel__right2,axiom,
    ! [A: complex,C2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2596_mult__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2597_mult__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2598_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2599_mult__cancel__right1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2600_mult__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2601_mult__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2602_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2603_mult__cancel__left2,axiom,
    ! [C2: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2604_mult__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2605_mult__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2606_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2607_mult__cancel__left1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2608_mult__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2609_mult__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2610_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2611_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2612_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2613_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2614_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2615_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(C2),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2616_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2617_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_2618_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2619_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_2620_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_2621_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2622_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2623_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2624_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2625_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2626_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(B),C2)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2627_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2628_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_2629_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2630_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_2631_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_2632_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2633_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2634_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2635_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2636_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( ( C2 = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = zero_zero_complex ) )
      & ( ( C2 != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2637_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ( C2 = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) = zero_zero_rat ) )
      & ( ( C2 != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2638_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ( C2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = zero_zero_real ) )
      & ( ( C2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2639_div__mult__mult1__if,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ( C2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = zero_zero_nat ) )
      & ( ( C2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_2640_div__mult__mult1__if,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ( C2 = zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = zero_zero_int ) )
      & ( ( C2 != zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_2641_div__mult__mult2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_2642_div__mult__mult2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_2643_div__mult__mult1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_2644_div__mult__mult1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_2645_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_2646_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_2647_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_2648_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_2649_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] : times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(A,B),numera1916890842035813515d_enat(V)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(V)),times_7803423173614009249d_enat(B,numera1916890842035813515d_enat(V))) ).

% distrib_right_numeral
tff(fact_2650_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_2651_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C2)) ).

% distrib_left_numeral
tff(fact_2652_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% distrib_left_numeral
tff(fact_2653_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C2)) ).

% distrib_left_numeral
tff(fact_2654_distrib__left__numeral,axiom,
    ! [V: num,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C2)) ).

% distrib_left_numeral
tff(fact_2655_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C2: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(B,C2)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),B),times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),C2)) ).

% distrib_left_numeral
tff(fact_2656_distrib__left__numeral,axiom,
    ! [V: num,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C2)) ).

% distrib_left_numeral
tff(fact_2657_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2658_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
    <=> ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2659_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_2660_mult__less__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% mult_less_cancel2
tff(fact_2661_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2662_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) ).

% mult_Suc_right
tff(fact_2663_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( K2 = 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,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) = zero_zero_nat ) )
      & ( ( K2 != 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,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_2664_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K2))
       => ( aa_int_int(divide_divide_int(K2),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2665_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L))
       => ( aa_int_int(divide_divide_int(K2),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2666_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2667_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2668_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2669_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2670_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) = A )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2671_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) = A )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)) ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2672_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) = A )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2673_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) = B ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2674_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)) = B ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2675_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) = B ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2676_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2677_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2678_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2679_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2680_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_2681_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2682_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2683_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_2684_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2685_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2686_div__mult__self4,axiom,
    ! [B: nat,C2: 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),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_2687_div__mult__self4,axiom,
    ! [B: int,C2: 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),C2)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_2688_div__mult__self3,axiom,
    ! [B: nat,C2: 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,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_2689_div__mult__self3,axiom,
    ! [B: int,C2: 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,C2),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_2690_div__mult__self2,axiom,
    ! [B: nat,A: nat,C2: 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),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_2691_div__mult__self2,axiom,
    ! [B: int,A: int,C2: 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),C2))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_2692_div__mult__self1,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_2693_div__mult__self1,axiom,
    ! [B: int,A: int,C2: 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,C2),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_2694_one__le__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_2695_mult__le__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% mult_le_cancel2
tff(fact_2696_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2697_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2)),N) = M2 ) ) ).

% div_mult_self1_is_m
tff(fact_2698_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),N) = M2 ) ) ).

% div_mult_self_is_m
tff(fact_2699_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% add1_zle_eq
tff(fact_2700_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_2701_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A2),B))) ) ) ).

% zdiv_mono1
tff(fact_2702_zdiv__mono2,axiom,
    ! [A: int,B2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B2))) ) ) ) ).

% zdiv_mono2
tff(fact_2703_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(divide_divide_int(I2),K2) = zero_zero_int )
    <=> ( ( K2 = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),I2)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_2704_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A2),B)),aa_int_int(divide_divide_int(A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_2705_zdiv__mono2__neg,axiom,
    ! [A: int,B2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B2)),aa_int_int(divide_divide_int(A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2706_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_2707_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K2),L)))
    <=> ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_2708_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(K2),L))) ) ) ).

% div_positive_int
tff(fact_2709_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_2710_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_2711_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(I2),K2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),I2)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_2712_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_2713_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_2714_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_2715_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_2716_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_2717_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_2718_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_2719_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(X),K2)),X)) ) ) ).

% int_div_less_self
tff(fact_2720_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_2721_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_2722_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_2723_int__gr__induct,axiom,
    ! [K2: int,I2: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),I2))
     => ( pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),I3))
             => ( pp(aa_int_bool(P2,I3))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P2,I2)) ) ) ) ).

% int_gr_induct
tff(fact_2724_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B)),C2) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2725_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2726_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C2) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2727_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2728_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2729_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B)),C2) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C2)) ).

% mult.assoc
tff(fact_2730_mult_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% mult.assoc
tff(fact_2731_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C2) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C2)) ).

% mult.assoc
tff(fact_2732_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.assoc
tff(fact_2733_mult_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% mult.assoc
tff(fact_2734_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_2735_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% mult.commute
tff(fact_2736_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_2737_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_2738_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_2739_mult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C2: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(times_times_complex(A),C2)) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C2)) ).

% mult.left_commute
tff(fact_2740_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% mult.left_commute
tff(fact_2741_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C2)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C2)) ).

% mult.left_commute
tff(fact_2742_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: 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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.left_commute
tff(fact_2743_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% mult.left_commute
tff(fact_2744_less__int__code_I1_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_int_code(1)
tff(fact_2745_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_2746_enat__less__induct,axiom,
    ! [P2: fun_Ex8563670519792538868t_bool,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M4: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M4),N3))
             => pp(aa_Ext2772979874139974381t_bool(P2,M4)) )
         => pp(aa_Ext2772979874139974381t_bool(P2,N3)) )
     => pp(aa_Ext2772979874139974381t_bool(P2,N)) ) ).

% enat_less_induct
tff(fact_2747_plus__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),zero_zero_int) = K2 ).

% plus_int_code(1)
tff(fact_2748_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_2749_less__eq__int__code_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% less_eq_int_code(1)
tff(fact_2750_mult__right__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(times_times_complex(B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2751_mult__right__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2752_mult__right__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C2) = aa_rat_rat(times_times_rat(B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2753_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2754_mult__right__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2755_mult__left__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C2),A) = aa_complex_complex(times_times_complex(C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2756_mult__left__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2757_mult__left__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C2),A) = aa_rat_rat(times_times_rat(C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2758_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2759_mult__left__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2760_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_2761_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2762_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2763_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_2764_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_2765_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_2766_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2767_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2768_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_2769_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_2770_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_2771_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2772_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2773_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_2774_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_2775_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_2776_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2777_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2778_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_2779_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_2780_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2781_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2782_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2783_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_2784_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_2785_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C2: complex,D: complex] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C2)),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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2786_crossproduct__noteq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2787_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2788_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2789_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2790_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,X: complex,Z: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Y)),aa_complex_complex(times_times_complex(X),Z)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W2),Z)),aa_complex_complex(times_times_complex(X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2791_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2792_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Y)),aa_rat_rat(times_times_rat(X),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Z)),aa_rat_rat(times_times_rat(X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2793_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2794_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2795_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B: complex,C2: 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)),C2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),E2)),C2) ).

% combine_common_factor
tff(fact_2796_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C2) ).

% combine_common_factor
tff(fact_2797_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E2)),C2) ).

% combine_common_factor
tff(fact_2798_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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)),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C2) ).

% combine_common_factor
tff(fact_2799_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C2: 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)),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C2) ).

% combine_common_factor
tff(fact_2800_distrib__right,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)) ).

% distrib_right
tff(fact_2801_distrib__right,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% distrib_right
tff(fact_2802_distrib__right,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ).

% distrib_right
tff(fact_2803_distrib__right,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% distrib_right
tff(fact_2804_distrib__right,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% distrib_right
tff(fact_2805_distrib__left,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B),C2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C2)) ).

% distrib_left
tff(fact_2806_distrib__left,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ).

% distrib_left
tff(fact_2807_distrib__left,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C2)) ).

% distrib_left
tff(fact_2808_distrib__left,axiom,
    ! [A: nat,B: nat,C2: 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),C2)) = 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),C2)) ).

% distrib_left
tff(fact_2809_distrib__left,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% distrib_left
tff(fact_2810_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2811_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2812_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2813_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2814_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2815_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B),C2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2816_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2817_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2818_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2819_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2820_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2821_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2822_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2823_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(Y),W2)) ).

% times_divide_times_eq
tff(fact_2824_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(Z),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),W2)) ).

% times_divide_times_eq
tff(fact_2825_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W2)) ).

% times_divide_times_eq
tff(fact_2826_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W2)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2827_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(Z),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),W2)),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2828_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),W2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_2829_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C2) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C2),B)) ).

% divide_divide_eq_left'
tff(fact_2830_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C2) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C2),B)) ).

% divide_divide_eq_left'
tff(fact_2831_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ).

% divide_divide_eq_left'
tff(fact_2832_Suc__mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N) )
    <=> ( M2 = N ) ) ).

% Suc_mult_cancel1
tff(fact_2833_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_2834_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N) )
    <=> ( ( K2 = zero_zero_nat )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2835_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J))) ) ).

% mult_le_mono2
tff(fact_2836_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2))) ) ).

% mult_le_mono1
tff(fact_2837_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_2838_le__square,axiom,
    ! [M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),M2))) ).

% le_square
tff(fact_2839_le__cube,axiom,
    ! [M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),M2)))) ).

% le_cube
tff(fact_2840_add__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) ).

% add_mult_distrib2
tff(fact_2841_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)) ).

% add_mult_distrib
tff(fact_2842_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),U)),K2) ).

% left_add_mult_distrib
tff(fact_2843_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_2844_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_2845_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)) ).

% nat_mult_max_right
tff(fact_2846_nat__mult__max__left,axiom,
    ! [M2: nat,N: 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,M2),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_2847_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2848_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2849_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2850_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2851_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2852_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2853_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2854_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2855_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2856_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2857_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2858_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2859_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_2860_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2861_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2862_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2863_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_2864_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2865_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2866_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2867_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_2868_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2869_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2870_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2871_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_2872_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_2873_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_2874_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2875_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_2876_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2877_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2878_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2879_mult__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_right_mono
tff(fact_2880_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))) ) ) ).

% mult_right_mono
tff(fact_2881_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))) ) ) ).

% mult_right_mono
tff(fact_2882_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_right_mono
tff(fact_2883_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_2884_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_2885_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_2886_mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_left_mono
tff(fact_2887_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))) ) ) ).

% mult_left_mono
tff(fact_2888_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% mult_left_mono
tff(fact_2889_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_left_mono
tff(fact_2890_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2891_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2892_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_2893_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2894_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2895_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2896_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ).

% split_mult_pos_le
tff(fact_2897_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ).

% split_mult_pos_le
tff(fact_2898_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_2899_zero__le__square,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A))) ).

% zero_le_square
tff(fact_2900_zero__le__square,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A))) ).

% zero_le_square
tff(fact_2901_zero__le__square,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_2902_mult__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2903_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2904_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2905_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2906_mult__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2907_mult__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2908_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2909_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2910_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2911_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2912_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2913_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2914_mult__less__cancel__right__disj,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2915_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2916_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2917_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2918_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2919_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2920_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_2921_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2922_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2923_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2924_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2925_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2926_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2927_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2928_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2929_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2930_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2931_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2932_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2933_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2934_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2935_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2936_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2937_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2938_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2939_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2940_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2941_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2942_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2943_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_2944_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2945_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2946_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2947_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos
tff(fact_2948_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2949_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2950_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_2951_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_2952_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_2953_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_2954_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_2955_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_pos_pos
tff(fact_2956_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_pos_pos
tff(fact_2957_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_2958_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_pos_pos
tff(fact_2959_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_2960_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_2961_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_2962_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_2963_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_2964_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_2965_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_2966_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_2967_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_2968_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_2969_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_2970_not__square__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_2971_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_2972_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_2973_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_neg_neg
tff(fact_2974_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_neg_neg
tff(fact_2975_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_neg_neg
tff(fact_2976_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C2: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C2)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2977_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C2: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C2)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2978_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C2: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C2)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2979_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C2: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2980_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C2: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2981_less__1__mult,axiom,
    ! [M2: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M2),N))) ) ) ).

% less_1_mult
tff(fact_2982_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M2),N))) ) ) ).

% less_1_mult
tff(fact_2983_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))) ) ) ).

% less_1_mult
tff(fact_2984_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M2),N))) ) ) ).

% less_1_mult
tff(fact_2985_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2986_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X),Y) = aa_rat_rat(divide_divide_rat(W2),Z) )
        <=> ( aa_rat_rat(times_times_rat(X),Z) = aa_rat_rat(times_times_rat(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2987_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2988_divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_2989_divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C2) = A )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_2990_divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = A )
    <=> ( ( ( C2 != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_2991_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C2) = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_2992_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C2) = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_2993_eq__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_2994_divide__eq__imp,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C2) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_2995_divide__eq__imp,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C2) )
       => ( aa_rat_rat(divide_divide_rat(B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_2996_divide__eq__imp,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_2997_eq__divide__imp,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C2) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_2998_eq__divide__imp,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C2) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_2999_eq__divide__imp,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_3000_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3001_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C2) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3002_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3003_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
      <=> ( aa_complex_complex(times_times_complex(A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3004_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C2) )
      <=> ( aa_rat_rat(times_times_rat(A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3005_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3006_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_3007_power__add,axiom,
    ! [A: complex,M2: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_3008_power__add,axiom,
    ! [A: real,M2: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_add
tff(fact_3009_power__add,axiom,
    ! [A: rat,M2: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_3010_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_3011_power__add,axiom,
    ! [A: int,M2: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_3012_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J))) ) ) ).

% mult_less_mono2
tff(fact_3013_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2))) ) ) ).

% mult_less_mono1
tff(fact_3014_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_3015_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N) )
      <=> ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3016_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_3017_mult__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) ).

% mult_Suc
tff(fact_3018_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3019_less__mult__imp__div__less,axiom,
    ! [M2: nat,I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),I2)) ) ).

% less_mult_imp_div_less
tff(fact_3020_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),N)),M2)) ).

% div_times_less_eq_dividend
tff(fact_3021_times__div__less__eq__dividend,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N))),M2)) ).

% times_div_less_eq_dividend
tff(fact_3022_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3023_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3024_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3025_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3026_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3027_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3028_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3029_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3030_mult__right__le__imp__le,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3031_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3032_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3033_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3034_mult__left__le__imp__le,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3035_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3036_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3037_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3038_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3039_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3040_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3041_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3042_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3043_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3044_mult__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_3045_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_3046_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_3047_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3048_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3049_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3050_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3051_mult__right__less__imp__less,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3052_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3053_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3054_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_3055_mult__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_3056_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_3057_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_3058_mult__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3059_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3060_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3061_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3062_mult__left__less__imp__less,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3063_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3064_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3065_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_3066_mult__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_3067_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_3068_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_3069_mult__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_3070_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_3071_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_3072_mult__left__le,axiom,
    ! [C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),A)) ) ) ).

% mult_left_le
tff(fact_3073_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),A)) ) ) ).

% mult_left_le
tff(fact_3074_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),A)) ) ) ).

% mult_left_le
tff(fact_3075_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),A)) ) ) ).

% mult_left_le
tff(fact_3076_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_3077_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_3078_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3079_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(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_3080_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_3081_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_3082_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_3083_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_3084_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_3085_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_3086_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3087_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3088_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3089_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3090_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3091_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3092_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3093_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3094_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3095_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_3096_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_3097_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_3098_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3099_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3100_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_3101_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_3102_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_3103_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_3104_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_3105_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_3106_pos__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_3107_pos__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_3108_pos__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% pos_divide_less_eq
tff(fact_3109_pos__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% pos_divide_less_eq
tff(fact_3110_neg__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% neg_less_divide_eq
tff(fact_3111_neg__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% neg_less_divide_eq
tff(fact_3112_neg__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_3113_neg__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_3114_less__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_3115_less__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_3116_divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_3117_divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_3118_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = numera6690914467698888265omplex(W2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3119_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C2) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3120_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = aa_num_real(numeral_numeral_real,W2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3121_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2) = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3122_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2) = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3123_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2) = B ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3124_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3125_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3126_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3127_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3128_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3129_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3130_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3131_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(X),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3132_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3133_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3134_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3135_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3136_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3137_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3138_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3139_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = 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_3140_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_3141_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_3142_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),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_3143_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_3144_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = B ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_3145_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_less_power_Suc
tff(fact_3146_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3147_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_less_power_Suc
tff(fact_3148_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3149_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_gt1_lemma
tff(fact_3150_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_gt1_lemma
tff(fact_3151_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))) ) ).

% power_gt1_lemma
tff(fact_3152_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(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),N)))) ) ).

% power_gt1_lemma
tff(fact_3153_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2))) ) ) ).

% n_less_n_mult_m
tff(fact_3154_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))) ) ) ).

% n_less_m_mult_n
tff(fact_3155_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))) ) ) ).

% one_less_mult
tff(fact_3156_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_3157_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Q2)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_3158_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_3159_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [R3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3160_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_3161_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_3162_mult__less__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3163_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3164_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3165_mult__less__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3166_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(times_times_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3167_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3168_mult__less__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3169_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C2),A)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3170_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3171_mult__less__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3172_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(times_times_rat(C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3173_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3174_mult__le__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3175_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3176_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3177_mult__le__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3178_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3179_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3180_mult__le__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3181_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3182_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3183_mult__le__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3184_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3185_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3186_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_3187_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_3188_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_3189_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_3190_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_3191_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_3192_pos__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_3193_pos__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_3194_pos__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% pos_divide_le_eq
tff(fact_3195_pos__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% pos_divide_le_eq
tff(fact_3196_neg__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% neg_le_divide_eq
tff(fact_3197_neg__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% neg_le_divide_eq
tff(fact_3198_neg__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_3199_neg__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_3200_divide__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B))) ) ) ) ).

% divide_left_mono
tff(fact_3201_divide__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A)),aa_rat_rat(divide_divide_rat(C2),B))) ) ) ) ).

% divide_left_mono
tff(fact_3202_le__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_3203_le__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_3204_divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_3205_divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_3206_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3207_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3208_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3209_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3210_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3211_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3212_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3213_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% power_Suc_less
tff(fact_3214_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% power_Suc_less
tff(fact_3215_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% power_Suc_less
tff(fact_3216_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).

% power_Suc_less
tff(fact_3217_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_3218_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_3219_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_3220_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_3221_mult__2,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Z) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2
tff(fact_3222_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_3223_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_3224_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_3225_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_3226_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_3227_mult__2__right,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(Z,numera1916890842035813515d_enat(aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2_right
tff(fact_3228_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_3229_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3230_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3231_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3232_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3233_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] : plus_p3455044024723400733d_enat(A,plus_p3455044024723400733d_enat(A,B)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),B) ).

% left_add_twice
tff(fact_3234_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3235_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q2))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_3236_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3237_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N))))) ) ).

% dividend_less_times_div
tff(fact_3238_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_3239_split__div,axiom,
    ! [P2: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P2,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P2,I4)) ) ) ) ) ) ).

% split_div
tff(fact_3240_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3241_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3242_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3243_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3244_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3245_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3246_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3247_sum__squares__bound,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3248_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3249_split__div_H,axiom,
    ! [P2: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P2,zero_zero_nat)) )
        | ? [Q5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q5)),M2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q5))))
            & pp(aa_nat_bool(P2,Q5)) ) ) ) ).

% split_div'
tff(fact_3250_power2__sum,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3251_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3252_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3253_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3254_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] : power_8040749407984259932d_enat(plus_p3455044024723400733d_enat(X,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(power_8040749407984259932d_enat(X,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8040749407984259932d_enat(Y,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_7803423173614009249d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),X),Y)) ).

% power2_sum
tff(fact_3255_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3256_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_3257_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_3258_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_3259_nat__bit__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))) ) )
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P2,N3))
             => pp(aa_nat_bool(P2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)))) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_bit_induct
tff(fact_3260_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3261_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(X),Y) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3262_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_3263_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),N) = A )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),N) = A ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3264_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_3265_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3266_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3267_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3268_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_3269_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_3270_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_3271_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3272_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3273_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3274_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3275_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3276_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X)),aa_rat_rat(times_times_rat(Z),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3277_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3278_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3279_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3280_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3281_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_3282_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,product_Pair_int_int(Q2,R2)))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_3283_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% low_def
tff(fact_3284_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3285_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3286_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),nat_set_decode(X)))
    <=> pp(aa_set_nat_bool(member_nat(N),nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% set_decode_Suc
tff(fact_3287_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B3) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y = vEBT_Leaf(A3,fTrue) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts2,S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2) )
                   => ( Y != if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3288_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_3289_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),one_one_nat))
    <=> ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3290_finite__Collect__disjI,axiom,
    ! [P2: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_af(P2),Q))))
    <=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(P2)))
        & pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3291_finite__Collect__disjI,axiom,
    ! [P2: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(P2),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P2))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3292_finite__Collect__disjI,axiom,
    ! [P2: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(P2),Q))))
    <=> ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P2)))
        & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3293_finite__Collect__disjI,axiom,
    ! [P2: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(P2),Q))))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
        & pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3294_finite__Collect__disjI,axiom,
    ! [P2: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_aj(P2),Q))))
    <=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(P2)))
        & pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3295_finite__Collect__disjI,axiom,
    ! [P2: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ak(P2),Q))))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P2)))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3296_finite__Collect__disjI,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_al(P2),Q))))
    <=> ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P2)))
        & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3297_finite__Collect__conjI,axiom,
    ! [P2: fun_real_bool,Q: fun_real_bool] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(P2)))
        | pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) )
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_am(P2),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3298_finite__Collect__conjI,axiom,
    ! [P2: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P2))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_an(P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_3299_finite__Collect__conjI,axiom,
    ! [P2: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P2)))
        | pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) )
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_ao(P2),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3300_finite__Collect__conjI,axiom,
    ! [P2: fun_nat_bool,Q: fun_nat_bool] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
        | pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) )
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ap(P2),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3301_finite__Collect__conjI,axiom,
    ! [P2: fun_int_bool,Q: fun_int_bool] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(P2)))
        | pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_aq(P2),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3302_finite__Collect__conjI,axiom,
    ! [P2: fun_complex_bool,Q: fun_complex_bool] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P2)))
        | pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ar(P2),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3303_finite__Collect__conjI,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P2)))
        | pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Q))) )
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun830836278467416944t_bool(aTP_Lamm_as(P2),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3304_finite__interval__int1,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_at(A),B)))) ).

% finite_interval_int1
tff(fact_3305_finite__interval__int4,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_au(A),B)))) ).

% finite_interval_int4
tff(fact_3306_dvd__0__right,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),zero_zero_complex)) ).

% dvd_0_right
tff(fact_3307_dvd__0__right,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),zero_zero_real)) ).

% dvd_0_right
tff(fact_3308_dvd__0__right,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),zero_zero_rat)) ).

% dvd_0_right
tff(fact_3309_dvd__0__right,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% dvd_0_right
tff(fact_3310_dvd__0__right,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),zero_zero_int)) ).

% dvd_0_right
tff(fact_3311_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_3312_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_3313_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_3314_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_3315_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_3316_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3317_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3318_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3319_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_3320_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3321_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3322_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3323_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_3324_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M2 = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_3325_dvd__1__left,axiom,
    ! [K2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K2)) ).

% dvd_1_left
tff(fact_3326_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_3327_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_3328_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_3329_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_3330_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_3331_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_3332_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_3333_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_3334_div__dvd__div,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),A)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ) ).

% div_dvd_div
tff(fact_3335_div__dvd__div,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(B),A)),aa_int_int(divide_divide_int(C2),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ) ).

% div_dvd_div
tff(fact_3336_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3337_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_3338_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_3339_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_3340_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_3341_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
    <=> ( ( K2 = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3342_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),K2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_3343_set__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),K2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_3344_mod__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = M2 ) ) ).

% mod_less
tff(fact_3345_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_av,A4)))) ) ).

% finite_Collect_subsets
tff(fact_3346_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_aw(A4))) ) ).

% finite_Collect_subsets
tff(fact_3347_finite__Collect__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_set7594618562247239262t_bool(aTP_Lamm_ax,A4))) ) ).

% finite_Collect_subsets
tff(fact_3348_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_ay(A4))) ) ).

% finite_Collect_subsets
tff(fact_3349_finite__interval__int3,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_az(A),B)))) ).

% finite_interval_int3
tff(fact_3350_finite__interval__int2,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ba(A),B)))) ).

% finite_interval_int2
tff(fact_3351_finite__Collect__less__nat,axiom,
    ! [K2: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,K2)))) ).

% finite_Collect_less_nat
tff(fact_3352_finite__Collect__le__nat,axiom,
    ! [K2: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bc,K2)))) ).

% finite_Collect_le_nat
tff(fact_3353_set__decode__inverse,axiom,
    ! [N: nat] : aa_set_nat_nat(nat_set_encode,nat_set_decode(N)) = N ).

% set_decode_inverse
tff(fact_3354_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3355_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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,C2),A)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3356_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3357_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3358_dvd__mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)))
    <=> ( ( C2 = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3359_dvd__mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( C2 = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3360_dvd__mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)))
    <=> ( ( C2 = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3361_dvd__mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( C2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3362_dvd__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)))
    <=> ( ( C2 = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3363_dvd__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( C2 = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3364_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(C2),A)),aa_rat_rat(times_times_rat(C2),B)))
    <=> ( ( C2 = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3365_dvd__mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( C2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3366_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ).

% unit_prod
tff(fact_3367_unit__prod,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ).

% unit_prod
tff(fact_3368_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(C2),A)),B)))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3369_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3370_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C2),A)),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3371_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),A)),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3372_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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,C2),A)),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3373_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(C2),A))))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3374_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3375_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C2),A))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3376_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),A))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3377_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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,C2),A))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3378_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_3379_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_3380_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_3381_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_3382_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_3383_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_3384_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_3385_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_3386_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3387_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3388_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3389_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3390_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3391_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3392_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3393_unit__div__1__unit,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_3394_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat)) ) ) ).

% unit_div
tff(fact_3395_unit__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int)) ) ) ).

% unit_div
tff(fact_3396_div__add,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ) ).

% div_add
tff(fact_3397_div__add,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C2)),aa_int_int(divide_divide_int(B),C2)) ) ) ) ).

% div_add
tff(fact_3398_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_3399_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_3400_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_3401_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_3402_mod__mult__self1,axiom,
    ! [A: nat,C2: 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,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_3403_mod__mult__self1,axiom,
    ! [A: int,C2: 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,C2),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_3404_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_3405_mod__mult__self2,axiom,
    ! [A: int,B: int,C2: 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),C2))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_3406_mod__mult__self3,axiom,
    ! [C2: 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,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_3407_mod__mult__self3,axiom,
    ! [C2: 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,C2),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_3408_mod__mult__self4,axiom,
    ! [B: nat,C2: 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),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_3409_mod__mult__self4,axiom,
    ! [B: int,C2: 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),C2)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_3410_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3411_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_3412_mod__by__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_3413_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_3414_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_3415_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3416_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3417_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3418_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3419_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_3420_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_3421_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K2: nat,N: 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,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self1
tff(fact_3422_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self2
tff(fact_3423_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)),M2))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self3
tff(fact_3424_Suc__mod__mult__self4,axiom,
    ! [N: nat,K2: nat,M2: 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,N),K2)),M2))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self4
tff(fact_3425_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_3426_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_3427_even__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_3428_even__add,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_3429_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),nat_set_decode(X)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)) ) ).

% set_decode_0
tff(fact_3430_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3431_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3432_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3433_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_3434_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq
tff(fact_3435_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_3436_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3437_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3438_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3439_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_3440_even__plus__one__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_3441_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3442_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3443_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3444_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3445_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_3446_add__self__mod__2,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),M2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_3447_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3448_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3449_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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 )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3450_even__succ__div__2,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3451_even__succ__div__2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3452_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3453_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3454_even__succ__div__two,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3455_even__succ__div__two,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3456_even__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_3457_even__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_3458_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_3459_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3460_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3461_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3462_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3463_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3464_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3465_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3466_int__ge__induct,axiom,
    ! [K2: int,I2: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),I2))
     => ( pp(aa_int_bool(P2,K2))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),I3))
             => ( pp(aa_int_bool(P2,I3))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P2,I2)) ) ) ) ).

% int_ge_induct
tff(fact_3467_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N3))) ) ).

% real_arch_pow
tff(fact_3468_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N3)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_3469_zdvd__mult__cancel,axiom,
    ! [K2: int,M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),M2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),N)))
     => ( ( K2 != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_3470_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 )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_3471_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_3472_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3473_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3474_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 )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_3475_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_3476_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero_int )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bd,I2)))) ) ).

% finite_divisors_int
tff(fact_3477_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_be,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_be,B))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3478_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf,B))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3479_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_bd,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_bd,B))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3480_zdvd__reduce,axiom,
    ! [K2: int,N: int,M2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,K2),M2))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),N)) ) ).

% zdvd_reduce
tff(fact_3481_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D))),T))) ) ) ).

% zdvd_period
tff(fact_3482_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_3483_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_3484_less__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A4),B4))
    <=> pp(aa_fun8663595511526100224l_bool(ord_le7158584875777748973t_bool(aTP_Lamm_a(A4)),aTP_Lamm_a(B4))) ) ).

% less_set_def
tff(fact_3485_less__set__def,axiom,
    ! [A4: set_real,B4: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A4),B4))
    <=> ord_le7519368973566061808l_bool(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4)) ) ).

% less_set_def
tff(fact_3486_less__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A4),B4))
    <=> ord_le4523237222650602006t_bool(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4)) ) ).

% less_set_def
tff(fact_3487_less__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A4),B4))
    <=> ord_le6841235830588351564t_bool(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_set_def
tff(fact_3488_less__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
    <=> ord_le3716944864119727344t_bool(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4)) ) ).

% less_set_def
tff(fact_3489_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_3490_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_3491_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_be,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_be,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% subset_divisors_dvd
tff(fact_3492_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_3493_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_bd,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_bd,B))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_3494_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_bg) ).

% empty_def
tff(fact_3495_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_bh) ).

% empty_def
tff(fact_3496_empty__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_bi) ).

% empty_def
tff(fact_3497_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_bj) ).

% empty_def
tff(fact_3498_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bk) ).

% empty_def
tff(fact_3499_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bl) ).

% empty_def
tff(fact_3500_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_nat,R: fun_re9121486324101951821t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B4))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B4))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bm(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3501_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_int,R: fun_re8661203372455305201t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B4))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B4))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bn(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3502_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_complex,R: fun_re3010880487650781423x_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B4))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B4))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bo(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3503_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_nat,R: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A4))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B4))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B4))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bp(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3504_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_int,R: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A4))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B4))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B4))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bq(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3505_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_complex,R: fun_na5941084329111710283x_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A4))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B4))
                  & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B4))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_br(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3506_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_nat,R: fun_int_fun_nat_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B4))
                  & pp(aa_nat_bool(aa_int_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B4))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bs(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3507_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_int,R: fun_int_fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B4))
                  & pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B4))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_bt(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3508_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_complex,R: fun_in5750378413879446255x_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B4))
                  & pp(aa_complex_bool(aa_int5354780257279585880x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B4))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_bu(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3509_pigeonhole__infinite__rel,axiom,
    ! [A4: set_complex,B4: set_nat,R: fun_co2841405903858895691t_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A4))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B4))
                  & pp(aa_nat_bool(aa_com5335841084339320244t_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B4))
              & ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bv(A4),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3510_not__finite__existsD,axiom,
    ! [P2: fun_real_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(P2)))
     => ? [X_12: real] : pp(aa_real_bool(P2,X_12)) ) ).

% not_finite_existsD
tff(fact_3511_not__finite__existsD,axiom,
    ! [P2: fun_list_nat_bool] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P2))
     => ? [X_12: list_nat] : pp(aa_list_nat_bool(P2,X_12)) ) ).

% not_finite_existsD
tff(fact_3512_not__finite__existsD,axiom,
    ! [P2: fun_set_nat_bool] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P2)))
     => ? [X_12: set_nat] : pp(aa_set_nat_bool(P2,X_12)) ) ).

% not_finite_existsD
tff(fact_3513_not__finite__existsD,axiom,
    ! [P2: fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => ? [X_12: nat] : pp(aa_nat_bool(P2,X_12)) ) ).

% not_finite_existsD
tff(fact_3514_not__finite__existsD,axiom,
    ! [P2: fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(P2)))
     => ? [X_12: int] : pp(aa_int_bool(P2,X_12)) ) ).

% not_finite_existsD
tff(fact_3515_not__finite__existsD,axiom,
    ! [P2: fun_complex_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P2)))
     => ? [X_12: complex] : pp(aa_complex_bool(P2,X_12)) ) ).

% not_finite_existsD
tff(fact_3516_not__finite__existsD,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,collec3392354462482085612at_nat(P2)))
     => ? [X_12: product_prod_nat_nat] : pp(aa_Pro2232559814647311250t_bool(P2,X_12)) ) ).

% not_finite_existsD
tff(fact_3517_dvd__mod__imp__dvd,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3518_dvd__mod__imp__dvd,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),aa_int_int(modulo_modulo_int(A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3519_dvd__mod__iff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A)) ) ) ).

% dvd_mod_iff
tff(fact_3520_dvd__mod__iff,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),aa_int_int(modulo_modulo_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A)) ) ) ).

% dvd_mod_iff
tff(fact_3521_dvd__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_trans
tff(fact_3522_dvd__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_trans
tff(fact_3523_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_3524_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_3525_pred__equals__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ! [X5: product_prod_nat_nat,Xa3: product_prod_nat_nat] :
          ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X5,Xa3),R)
        <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(X5,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3526_pred__equals__eq2,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ! [X5: set_Pr4329608150637261639at_nat,Xa3: set_Pr4329608150637261639at_nat] :
          ( member1466754251312161552at_nat(produc9060074326276436823at_nat(X5,Xa3),R)
        <=> member1466754251312161552at_nat(produc9060074326276436823at_nat(X5,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3527_pred__equals__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Xa3: set_Pr1261947904930325089at_nat] :
          ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Xa3)),R))
        <=> pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Xa3)),S2)) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3528_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Xa3: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa3)),R))
        <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa3)),S2)) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3529_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Xa3: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X5,Xa3),R)
        <=> member5262025264175285858nt_int(product_Pair_int_int(X5,Xa3),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3530_lambda__zero,axiom,
    aTP_Lamm_bw = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_3531_lambda__zero,axiom,
    aTP_Lamm_bx = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_3532_lambda__zero,axiom,
    aTP_Lamm_by = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_3533_lambda__zero,axiom,
    aTP_Lamm_bz = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3534_lambda__zero,axiom,
    aTP_Lamm_ca = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3535_lambda__one,axiom,
    aTP_Lamm_cb = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_3536_lambda__one,axiom,
    aTP_Lamm_cc = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_3537_lambda__one,axiom,
    aTP_Lamm_cd = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_3538_lambda__one,axiom,
    aTP_Lamm_ce = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_3539_lambda__one,axiom,
    aTP_Lamm_cf = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_3540_max__def__raw,axiom,
    ! [X3: set_int,Xa: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Xa))
       => ( ord_max_set_int(X3,Xa) = Xa ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(X3),Xa))
       => ( ord_max_set_int(X3,Xa) = X3 ) ) ) ).

% max_def_raw
tff(fact_3541_max__def__raw,axiom,
    ! [X3: rat,Xa: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Xa))
       => ( aa_rat_rat(ord_max_rat(X3),Xa) = Xa ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Xa))
       => ( aa_rat_rat(ord_max_rat(X3),Xa) = X3 ) ) ) ).

% max_def_raw
tff(fact_3542_max__def__raw,axiom,
    ! [X3: num,Xa: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Xa))
       => ( aa_num_num(ord_max_num(X3),Xa) = Xa ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Xa))
       => ( aa_num_num(ord_max_num(X3),Xa) = X3 ) ) ) ).

% max_def_raw
tff(fact_3543_max__def__raw,axiom,
    ! [X3: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Xa) = X3 ) ) ) ).

% max_def_raw
tff(fact_3544_max__def__raw,axiom,
    ! [X3: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Xa) = X3 ) ) ) ).

% max_def_raw
tff(fact_3545_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf,M2)))) ) ).

% finite_divisors_nat
tff(fact_3546_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ord_le2087689725136068685t_bool(aTP_Lamm_cg(R),aTP_Lamm_cg(S2))
    <=> ord_le3000389064537975527at_nat(R,S2) ) ).

% pred_subset_eq2
tff(fact_3547_pred__subset__eq2,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ord_le3287501503639976013t_bool(aTP_Lamm_ch(R),aTP_Lamm_ch(S2))
    <=> ord_le5997549366648089703at_nat(R,S2) ) ).

% pred_subset_eq2
tff(fact_3548_pred__subset__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ord_le1428200584844665933t_bool(aTP_Lamm_ci(R),aTP_Lamm_ci(S2))
    <=> pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,R),S2)) ) ).

% pred_subset_eq2
tff(fact_3549_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ord_le4899916226251588169t_bool(aTP_Lamm_cj(R),aTP_Lamm_cj(S2))
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,R),S2)) ) ).

% pred_subset_eq2
tff(fact_3550_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ord_le4546832483993983377t_bool(aTP_Lamm_ck(R),aTP_Lamm_ck(S2))
    <=> ord_le2843351958646193337nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_3551_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3552_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3553_bot__empty__eq2,axiom,
    ! [X3: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(bot_bo5325672116769796377t_bool,X3),Xa))
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Xa),bot_bo5327735625951526323at_nat) ) ).

% bot_empty_eq2
tff(fact_3554_bot__empty__eq2,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Xa: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(bot_bo4620695493499315481t_bool,X3),Xa))
    <=> member1466754251312161552at_nat(produc9060074326276436823at_nat(X3,Xa),bot_bo4948859079157340979at_nat) ) ).

% bot_empty_eq2
tff(fact_3555_bot__empty__eq2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(bot_bo3728952843091432729t_bool,X3),Xa))
    <=> pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X3,Xa)),bot_bo228742789529271731at_nat)) ) ).

% bot_empty_eq2
tff(fact_3556_bot__empty__eq2,axiom,
    ! [X3: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(bot_bo6207810721336151133t_bool,X3),Xa))
    <=> member5262025264175285858nt_int(product_Pair_int_int(X3,Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_3557_bot__empty__eq2,axiom,
    ! [X3: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bot_bo6560894463593755925t_bool,X3),Xa))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa)),bot_bo2099793752762293965at_nat)) ) ).

% bot_empty_eq2
tff(fact_3558_finite__M__bounded__by__nat,axiom,
    ! [P2: fun_nat_bool,I2: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cl(P2),I2)))) ).

% finite_M_bounded_by_nat
tff(fact_3559_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),aa_nat_nat(F,N3)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cm(F),U)))) ) ).

% finite_less_ub
tff(fact_3560_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3561_mod__add__eq,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_eq
tff(fact_3562_mod__add__eq,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(modulo_modulo_int(B),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_eq
tff(fact_3563_mod__add__cong,axiom,
    ! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ) ) ) ).

% mod_add_cong
tff(fact_3564_mod__add__cong,axiom,
    ! [A: int,C2: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C2) = aa_int_int(modulo_modulo_int(A2),C2) )
     => ( ( aa_int_int(modulo_modulo_int(B),C2) = aa_int_int(modulo_modulo_int(B2),C2) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) ) ) ) ).

% mod_add_cong
tff(fact_3565_mod__add__left__eq,axiom,
    ! [A: nat,C2: 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),C2)),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_left_eq
tff(fact_3566_mod__add__left__eq,axiom,
    ! [A: int,C2: 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),C2)),B)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_left_eq
tff(fact_3567_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_right_eq
tff(fact_3568_mod__add__right__eq,axiom,
    ! [A: int,B: int,C2: 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),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_right_eq
tff(fact_3569_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_3570_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_3571_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_3572_dvd__0__left,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_3573_dvd__0__left,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_3574_dvd__0__left,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_3575_dvd__0__left,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_3576_dvd__0__left,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_3577_dvd__triv__right,axiom,
    ! [A: complex,B: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B),A))) ).

% dvd_triv_right
tff(fact_3578_dvd__triv__right,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))) ).

% dvd_triv_right
tff(fact_3579_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A))) ).

% dvd_triv_right
tff(fact_3580_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))) ).

% dvd_triv_right
tff(fact_3581_dvd__triv__right,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))) ).

% dvd_triv_right
tff(fact_3582_dvd__mult__right,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),B)),C2))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),C2)) ) ).

% dvd_mult_right
tff(fact_3583_dvd__mult__right,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C2)) ) ).

% dvd_mult_right
tff(fact_3584_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C2)) ) ).

% dvd_mult_right
tff(fact_3585_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ).

% dvd_mult_right
tff(fact_3586_dvd__mult__right,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ).

% dvd_mult_right
tff(fact_3587_mult__dvd__mono,axiom,
    ! [A: complex,B: complex,C2: complex,D: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),D))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3588_mult__dvd__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3589_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3590_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3591_mult__dvd__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3592_dvd__triv__left,axiom,
    ! [A: complex,B: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(A),B))) ).

% dvd_triv_left
tff(fact_3593_dvd__triv__left,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ).

% dvd_triv_left
tff(fact_3594_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B))) ).

% dvd_triv_left
tff(fact_3595_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ).

% dvd_triv_left
tff(fact_3596_dvd__triv__left,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ).

% dvd_triv_left
tff(fact_3597_dvd__mult__left,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),B)),C2))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),C2)) ) ).

% dvd_mult_left
tff(fact_3598_dvd__mult__left,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2)) ) ).

% dvd_mult_left
tff(fact_3599_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2)) ) ).

% dvd_mult_left
tff(fact_3600_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ).

% dvd_mult_left
tff(fact_3601_dvd__mult__left,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ).

% dvd_mult_left
tff(fact_3602_dvd__mult2,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B),C2))) ) ).

% dvd_mult2
tff(fact_3603_dvd__mult2,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ).

% dvd_mult2
tff(fact_3604_dvd__mult2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C2))) ) ).

% dvd_mult2
tff(fact_3605_dvd__mult2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))) ) ).

% dvd_mult2
tff(fact_3606_dvd__mult2,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ).

% dvd_mult2
tff(fact_3607_dvd__mult,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),C2))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),aa_complex_complex(times_times_complex(B),C2))) ) ).

% dvd_mult
tff(fact_3608_dvd__mult,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ).

% dvd_mult
tff(fact_3609_dvd__mult,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C2))) ) ).

% dvd_mult
tff(fact_3610_dvd__mult,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))) ) ).

% dvd_mult
tff(fact_3611_dvd__mult,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ).

% dvd_mult
tff(fact_3612_dvd__def,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
    <=> ? [K3: complex] : A = aa_complex_complex(times_times_complex(B),K3) ) ).

% dvd_def
tff(fact_3613_dvd__def,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_3614_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_3615_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_3616_dvd__def,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_3617_dvdI,axiom,
    ! [A: complex,B: complex,K2: complex] :
      ( ( A = aa_complex_complex(times_times_complex(B),K2) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A)) ) ).

% dvdI
tff(fact_3618_dvdI,axiom,
    ! [A: real,B: real,K2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).

% dvdI
tff(fact_3619_dvdI,axiom,
    ! [A: rat,B: rat,K2: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K2) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).

% dvdI
tff(fact_3620_dvdI,axiom,
    ! [A: nat,B: nat,K2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% dvdI
tff(fact_3621_dvdI,axiom,
    ! [A: int,B: int,K2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% dvdI
tff(fact_3622_dvdE,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ~ ! [K: complex] : A != aa_complex_complex(times_times_complex(B),K) ) ).

% dvdE
tff(fact_3623_dvdE,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ~ ! [K: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) ) ).

% dvdE
tff(fact_3624_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ~ ! [K: rat] : A != aa_rat_rat(times_times_rat(B),K) ) ).

% dvdE
tff(fact_3625_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ~ ! [K: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) ) ).

% dvdE
tff(fact_3626_dvdE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ~ ! [K: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K) ) ).

% dvdE
tff(fact_3627_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).

% dvd_unit_imp_unit
tff(fact_3628_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int)) ) ) ).

% dvd_unit_imp_unit
tff(fact_3629_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% unit_imp_dvd
tff(fact_3630_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% unit_imp_dvd
tff(fact_3631_one__dvd,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A)) ).

% one_dvd
tff(fact_3632_one__dvd,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A)) ).

% one_dvd
tff(fact_3633_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_3634_one__dvd,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,one_one_complex),A)) ).

% one_dvd
tff(fact_3635_one__dvd,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,one_one_rat),A)) ).

% one_dvd
tff(fact_3636_dvd__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))) ) ) ).

% dvd_add
tff(fact_3637_dvd__add,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))) ) ) ).

% dvd_add
tff(fact_3638_dvd__add,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))) ) ) ).

% dvd_add
tff(fact_3639_dvd__add,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C2))) ) ) ).

% dvd_add
tff(fact_3640_dvd__add__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3641_dvd__add__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3642_dvd__add__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3643_dvd__add__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3644_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_3645_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_3646_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_3647_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_3648_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3649_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_3650_dvd__div__eq__cancel,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3651_dvd__div__eq__cancel,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),C2) = aa_int_int(divide_divide_int(B),C2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3652_dvd__div__eq__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3653_dvd__div__eq__iff,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3654_dvd__div__eq__iff,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( ( aa_int_int(divide_divide_int(A),C2) = aa_int_int(divide_divide_int(B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3655_dvd__div__eq__iff,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),B))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3656_mod__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)),M2)) ).

% mod_less_eq_dividend
tff(fact_3657_gcd__nat_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_3658_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_3659_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_3660_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_3661_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_3662_zdvd__antisym__nonneg,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M2))
           => ( M2 = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3663_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cn,X)) ).

% set_decode_def
tff(fact_3664_times__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K2),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3665_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_3666_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_code(2)
tff(fact_3667_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_code(2)
tff(fact_3668_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_code(2)
tff(fact_3669_numeral__code_I2_J,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit0,N)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)) ).

% numeral_code(2)
tff(fact_3670_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_code(2)
tff(fact_3671_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_3672_enat__0__less__mult__iff,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M2,N)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M2))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_3673_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3674_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3675_subset__decode__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,nat_set_decode(M2)),nat_set_decode(N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% subset_decode_imp_le
tff(fact_3676_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3677_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3678_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3679_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3680_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_3681_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_3682_mod__eqE,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C2) = aa_int_int(modulo_modulo_int(B),C2) )
     => ~ ! [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,C2),D4)) ) ).

% mod_eqE
tff(fact_3683_div__add1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2)) ).

% div_add1_eq
tff(fact_3684_div__add1__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = 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),C2)),aa_int_int(divide_divide_int(B),C2))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C2)),aa_int_int(modulo_modulo_int(B),C2))),C2)) ).

% div_add1_eq
tff(fact_3685_not__is__unit__0,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),one_one_nat)) ).

% not_is_unit_0
tff(fact_3686_not__is__unit__0,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),one_one_int)) ).

% not_is_unit_0
tff(fact_3687_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% minf(10)
tff(fact_3688_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S))) ) ) ).

% minf(10)
tff(fact_3689_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% minf(10)
tff(fact_3690_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))) ) ) ).

% minf(10)
tff(fact_3691_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% minf(9)
tff(fact_3692_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S))) ) ) ).

% minf(9)
tff(fact_3693_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% minf(9)
tff(fact_3694_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))) ) ) ).

% minf(9)
tff(fact_3695_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% pinf(10)
tff(fact_3696_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S))) ) ) ).

% pinf(10)
tff(fact_3697_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% pinf(10)
tff(fact_3698_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))) ) ) ).

% pinf(10)
tff(fact_3699_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% pinf(9)
tff(fact_3700_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S))) ) ) ).

% pinf(9)
tff(fact_3701_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% pinf(9)
tff(fact_3702_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))) ) ) ).

% pinf(9)
tff(fact_3703_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(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_3704_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3705_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3706_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3707_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3708_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat)) ) ) ).

% is_unit_mult_iff
tff(fact_3709_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int)) ) ) ).

% is_unit_mult_iff
tff(fact_3710_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_mult_unit_iff
tff(fact_3711_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_mult_unit_iff
tff(fact_3712_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% mult_unit_dvd_iff
tff(fact_3713_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% mult_unit_dvd_iff
tff(fact_3714_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3715_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3716_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3717_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3718_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),C2) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_3719_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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),C2) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_3720_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_3721_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_3722_mod__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ) ) ) ).

% mod_Suc
tff(fact_3723_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,nat_set_decode(N))) ).

% finite_set_decode
tff(fact_3724_mod__induct,axiom,
    ! [P2: fun_nat_bool,N: nat,P: nat,M2: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),P))
         => ( ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),P))
               => ( pp(aa_nat_bool(P2,N3))
                 => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N3)),P))) ) )
           => pp(aa_nat_bool(P2,M2)) ) ) ) ) ).

% mod_induct
tff(fact_3725_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),C2))
       => ( 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,C2),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3726_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),C2))
       => ( 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(C2),D)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3727_dvd__mult__imp__div,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2))) ) ).

% dvd_mult_imp_div
tff(fact_3728_dvd__mult__imp__div,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C2))) ) ).

% dvd_mult_imp_div
tff(fact_3729_dvd__div__mult2__eq,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_3730_dvd__div__mult2__eq,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)),A))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_3731_div__div__eq__right,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_3732_div__div__eq__right,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_3733_div__mult__swap,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) ) ) ).

% div_mult_swap
tff(fact_3734_div__mult__swap,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C2)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) ) ) ).

% div_mult_swap
tff(fact_3735_dvd__div__mult,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)),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)),C2) ) ) ).

% dvd_div_mult
tff(fact_3736_dvd__div__mult,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),C2)),A) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C2) ) ) ).

% dvd_div_mult
tff(fact_3737_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)),N)) ) ).

% mod_less_divisor
tff(fact_3738_gcd__nat__induct,axiom,
    ! [P2: fun_nat_fun_nat_bool,M2: nat,N: nat] :
      ( ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M),zero_zero_nat))
     => ( ! [M: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,N3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N3)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M),N3)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M2),N)) ) ) ).

% gcd_nat_induct
tff(fact_3739_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_div_cancel
tff(fact_3740_unit__div__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_div_cancel
tff(fact_3741_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% div_unit_dvd_iff
tff(fact_3742_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% div_unit_dvd_iff
tff(fact_3743_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_div_unit_iff
tff(fact_3744_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_div_unit_iff
tff(fact_3745_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3746_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C2)),aa_int_int(divide_divide_int(B),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3747_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3748_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C2)),aa_int_int(divide_divide_int(B),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3749_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_3750_mod__eq__0D,axiom,
    ! [M2: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),D) = zero_zero_nat )
     => ? [Q3: nat] : M2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_3751_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M2))) ) ) ).

% dvd_power_le
tff(fact_3752_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M2))) ) ) ).

% dvd_power_le
tff(fact_3753_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M2: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),M2))) ) ) ).

% dvd_power_le
tff(fact_3754_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),M2))) ) ) ).

% dvd_power_le
tff(fact_3755_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_3756_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_3757_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M2: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_3758_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_3759_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% le_imp_power_dvd
tff(fact_3760_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% le_imp_power_dvd
tff(fact_3761_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3762_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3763_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3764_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)) ) ) ).

% nat_dvd_not_less
tff(fact_3765_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2)) ) ) ).

% dvd_pos_nat
tff(fact_3766_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_3767_zdvd__not__zless,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M2),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M2)) ) ) ).

% zdvd_not_zless
tff(fact_3768_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3769_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X4: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3770_q__pos__lemma,axiom,
    ! [B2: int,Q6: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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,B2),Q6)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_3771_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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,B2),Q6)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3772_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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,B2),Q6)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3773_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3774_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3775_incr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_bool,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P2,X4))
           => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
         => ! [X3: int] :
              ( pp(aa_int_bool(P2,X3))
             => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3776_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),J))) ) ) ).

% zmult_zless_mono2
tff(fact_3777_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M2))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
      <=> ( ( M2 = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3778_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ) ).

% mod2_eq_if
tff(fact_3779_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_3780_parity__cases,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_3781_parity__cases,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3782_finite__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_co(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3783_finite__lists__length__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_cp(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3784_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cq(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3785_finite__lists__length__eq,axiom,
    ! [A4: set_bool,N: nat] :
      ( finite_finite_bool(A4)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cr(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3786_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cs(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3787_finite__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ct(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3788_even__set__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3789_even__set__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M2),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3790_even__unset__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M2),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3791_even__unset__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M2),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3792_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_3793_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3794_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3795_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3796_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_3797_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3798_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3799_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_3800_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_3801_finite__lists__length__le,axiom,
    ! [A4: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_cu(A4),N))) ) ).

% finite_lists_length_le
tff(fact_3802_finite__lists__length__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_cv(A4),N))) ) ).

% finite_lists_length_le
tff(fact_3803_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cw(A4),N))) ) ).

% finite_lists_length_le
tff(fact_3804_finite__lists__length__le,axiom,
    ! [A4: set_bool,N: nat] :
      ( finite_finite_bool(A4)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cx(A4),N))) ) ).

% finite_lists_length_le
tff(fact_3805_finite__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cy(A4),N))) ) ).

% finite_lists_length_le
tff(fact_3806_finite__lists__length__le,axiom,
    ! [A4: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_cz(A4),N))) ) ).

% finite_lists_length_le
tff(fact_3807_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = 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),C2))),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),C2))),C2)) ).

% div_mult1_eq
tff(fact_3808_div__mult1__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = 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),C2))),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),C2))),C2)) ).

% div_mult1_eq
tff(fact_3809_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_3810_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_3811_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_3812_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_3813_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_3814_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_3815_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_3816_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_3817_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_3818_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_3819_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_3820_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,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))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_3821_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_3822_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,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))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_3823_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) ) ) ).

% unit_dvdE
tff(fact_3824_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [C: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C) ) ) ).

% unit_dvdE
tff(fact_3825_unity__coeff__ex,axiom,
    ! [P2: fun_complex_bool,L: complex] :
      ( ? [X5: complex] : pp(aa_complex_bool(P2,aa_complex_complex(times_times_complex(L),X5)))
    <=> ? [X5: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X5),zero_zero_complex)))
          & pp(aa_complex_bool(P2,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3826_unity__coeff__ex,axiom,
    ! [P2: fun_real_bool,L: real] :
      ( ? [X5: real] : pp(aa_real_bool(P2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X5)))
    <=> ? [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),zero_zero_real)))
          & pp(aa_real_bool(P2,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3827_unity__coeff__ex,axiom,
    ! [P2: fun_rat_bool,L: rat] :
      ( ? [X5: rat] : pp(aa_rat_bool(P2,aa_rat_rat(times_times_rat(L),X5)))
    <=> ? [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X5),zero_zero_rat)))
          & pp(aa_rat_bool(P2,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3828_unity__coeff__ex,axiom,
    ! [P2: fun_nat_bool,L: nat] :
      ( ? [X5: nat] : pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X5)))
    <=> ? [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),zero_zero_nat)))
          & pp(aa_nat_bool(P2,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3829_unity__coeff__ex,axiom,
    ! [P2: fun_int_bool,L: int] :
      ( ? [X5: int] : pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(times_times_int,L),X5)))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),zero_zero_int)))
          & pp(aa_int_bool(P2,X5)) ) ) ).

% unity_coeff_ex
tff(fact_3830_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3831_dvd__div__div__eq__mult,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),D))
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D),C2) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3832_dvd__div__iff__mult,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( C2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3833_dvd__div__iff__mult,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( C2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C2)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3834_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3835_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C2))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3836_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3837_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
       => ( ( aa_int_int(divide_divide_int(B),A) = C2 )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3838_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3839_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_3840_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C2 )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) ) ) ) ).

% unit_eq_div1
tff(fact_3841_unit__eq__div1,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(divide_divide_int(A),B) = C2 )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) ) ) ) ).

% unit_eq_div1
tff(fact_3842_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_3843_unit__eq__div2,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( A = aa_int_int(divide_divide_int(C2),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_3844_div__mult__unit2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_3845_div__mult__unit2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_3846_unit__div__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),B) ) ) ).

% unit_div_commute
tff(fact_3847_unit__div__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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)),C2) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),B) ) ) ).

% unit_div_commute
tff(fact_3848_unit__div__mult__swap,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_3849_unit__div__mult__swap,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C2)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_3850_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3851_is__unit__div__mult2__eq,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3852_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)),N)) ) ).

% mod_le_divisor
tff(fact_3853_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3854_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3855_div__less__mono,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),N))) ) ) ) ) ).

% div_less_mono
tff(fact_3856_mod__eq__nat1E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ~ ! [S3: nat] : M2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_3857_mod__eq__nat2E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ~ ! [S3: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_3858_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q3: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3859_dvd__imp__le,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N)) ) ) ).

% dvd_imp_le
tff(fact_3860_mod__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ).

% mod_mult2_eq
tff(fact_3861_div__mod__decomp,axiom,
    ! [A4: nat,N: nat] : A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),N)) ).

% div_mod_decomp
tff(fact_3862_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3863_dvd__mult__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% dvd_mult_cancel
tff(fact_3864_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X4: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D4),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3865_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(divide_divide_int(A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3866_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(divide_divide_int(A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3867_split__zdiv,axiom,
    ! [P2: fun_int_bool,N: int,K2: int] :
      ( pp(aa_int_bool(P2,aa_int_int(divide_divide_int(N),K2)))
    <=> ( ( ( K2 = zero_zero_int )
         => pp(aa_int_bool(P2,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K2))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I4)),J3) ) )
             => pp(aa_int_bool(P2,I4)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I4)),J3) ) )
             => pp(aa_int_bool(P2,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_3868_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3869_even__zero,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat)) ).

% even_zero
tff(fact_3870_even__zero,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int)) ).

% even_zero
tff(fact_3871_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3872_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_3873_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3874_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_3875_is__unitE,axiom,
    ! [A: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3876_is__unitE,axiom,
    ! [A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(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(C2),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3877_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% odd_even_add
tff(fact_3878_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% odd_even_add
tff(fact_3879_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ) ).

% dvd_power_iff
tff(fact_3880_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M2)),aa_nat_int(power_power_int(X),N)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ) ).

% dvd_power_iff
tff(fact_3881_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_rat ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N))) ) ).

% dvd_power
tff(fact_3882_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))) ) ).

% dvd_power
tff(fact_3883_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ).

% dvd_power
tff(fact_3884_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),N))) ) ).

% dvd_power
tff(fact_3885_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N))) ) ).

% dvd_power
tff(fact_3886_split__mod,axiom,
    ! [P2: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P2,M2)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => pp(aa_nat_bool(P2,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_3887_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_3888_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),M2))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3889_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2)),M2))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3890_power__dvd__imp__le,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_dvd_imp_le
tff(fact_3891_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3892_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3893_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3894_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3895_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = 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)),C2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3896_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = 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)),C2))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3897_power__mono__odd,axiom,
    ! [N: nat,A: real,B: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono_odd
tff(fact_3898_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono_odd
tff(fact_3899_power__mono__odd,axiom,
    ! [N: nat,A: int,B: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono_odd
tff(fact_3900_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))),M2) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3901_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_3902_dvd__power__iff__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% dvd_power_iff_le
tff(fact_3903_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3904_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3905_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_3906_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_3907_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3908_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_3909_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M2: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_3910_oddE,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_3911_oddE,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_3912_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_le_even_power
tff(fact_3913_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_even_power
tff(fact_3914_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_even_power
tff(fact_3915_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% zero_le_odd_power
tff(fact_3916_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% zero_le_odd_power
tff(fact_3917_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_odd_power
tff(fact_3918_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3919_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3920_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3921_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),N))) ) ) ).

% verit_le_mono_div
tff(fact_3922_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd2)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3923_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A4)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) ) ) ).

% even_set_encode_iff
tff(fact_3924_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3925_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3926_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_3927_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc)),X))
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3928_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_3929_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_3930_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_3931_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3932_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3933_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3934_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_3935_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) = aa_int_int(modulo_modulo_int(X),M2) )
          | ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(X),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_3936_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3937_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3938_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3939_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3940_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                               => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3941_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3942_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3943_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X) = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)) ).

% vebt_insert.simps(5)
tff(fact_3944_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3945_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3946_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3947_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3948_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_3949_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_3950_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_3951_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_da(N),C2)))) ) ).

% finite_nth_roots
tff(fact_3952_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_db,N)))) ) ).

% finite_roots_unity
tff(fact_3953_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_dc,N)))) ) ).

% finite_roots_unity
tff(fact_3954_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B3) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y = vEBT_Leaf(A3,fTrue) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S3),Xa2)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3),Xa2)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2) )
                     => ( ( Y = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3955_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3956_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3957_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_3958_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys),N) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_3959_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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),N) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_3960_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_3961_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys),N) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_3962_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr2856333732135533678l_bool(product_bool_bool(Xs,Ys),N) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_3963_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_nat(Ys))))
     => ( nth_Pr2186422209488468047ol_nat(product_bool_nat(Xs,Ys),N) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_3964_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys))))
     => ( nth_Pr7231943226834047147ol_int(product_bool_int(Xs,Ys),N) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_3965_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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),N) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_3966_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr396318242284072693t_bool(product_nat_bool(Xs,Ys),N) = produc3913204875133826030t_bool(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_3967_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dd(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dd(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_de(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3968_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_df(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_df(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_dg(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3969_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dh(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dh(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_di(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3970_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dj(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dj(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_dk(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3971_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dl(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dl(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_dm(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3972_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dn(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dn(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_do(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3973_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_dp(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_dp(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_dq(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3974_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dr(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dr(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_ds(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3975_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_dt(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_dt(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_du(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3976_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_dv(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_dv(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_dw(I6),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_3977_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dx(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dx(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_dy(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3978_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_dz(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_dz(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_ea(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3979_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eb(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_eb(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ec(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3980_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ed(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ed(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ee(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3981_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_ef(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_ef(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_eg(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3982_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_eh(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_eh(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_ei(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3983_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ej(I6),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ej(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ek(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3984_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_el(I6),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_el(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_em(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3985_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_en(I6),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_en(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_eo(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3986_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ep(I6),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ep(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_eq(I6),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_3987_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                         => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3988_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3989_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N),K2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_3990_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K2))
       => ( aa_int_int(modulo_modulo_int(K2),L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_3991_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L))
       => ( aa_int_int(modulo_modulo_int(K2),L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_3992_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_3993_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool] : size_s1343096544515248124T_bool(produc5466675335756156456T_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_3994_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_3995_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_3996_length__product,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT] : size_s7211631430880329078T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_3997_length__product,axiom,
    ! [Xs: list_bool,Ys: list_bool] : size_s6988337448923680857l_bool(product_bool_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_3998_length__product,axiom,
    ! [Xs: list_bool,Ys: list_nat] : size_s3243772863075988794ol_nat(product_bool_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_3999_length__product,axiom,
    ! [Xs: list_bool,Ys: list_int] : size_s753689718144262038ol_int(product_bool_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4000_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_4001_length__product,axiom,
    ! [Xs: list_nat,Ys: list_bool] : size_s783543119881238752t_bool(product_nat_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_4002_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2))
       => ( M2 = N ) ) ) ).

% dvd_antisym
tff(fact_4003_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(K2),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_4004_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(modulo_modulo_int(K2),L))) ) ).

% neg_mod_bound
tff(fact_4005_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(K2),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_4006_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K2),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_4007_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(modulo_modulo_int(I2),K2) = I2 )
    <=> ( ( K2 = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),I2)) ) ) ) ).

% zmod_trivial_iff
tff(fact_4008_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ) ).

% pos_mod_conj
tff(fact_4009_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B))) ) ) ).

% neg_mod_conj
tff(fact_4010_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K2),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_4011_zdiv__mono__strict,axiom,
    ! [A4: int,B4: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A4),B4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => ( ( aa_int_int(modulo_modulo_int(A4),N) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B4),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A4),N)),aa_int_int(divide_divide_int(B4),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_4012_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int))
       => ( aa_int_int(modulo_modulo_int(K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_4013_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4014_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4015_split__zmod,axiom,
    ! [P2: fun_int_bool,N: int,K2: int] :
      ( pp(aa_int_bool(P2,aa_int_int(modulo_modulo_int(N),K2)))
    <=> ( ( ( K2 = zero_zero_int )
         => pp(aa_int_bool(P2,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K2))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I4)),J3) ) )
             => pp(aa_int_bool(P2,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
         => ! [I4: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I4)),J3) ) )
             => pp(aa_int_bool(P2,J3)) ) ) ) ) ).

% split_zmod
tff(fact_4016_split__neg__lemma,axiom,
    ! [K2: int,P2: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P2,aa_int_int(divide_divide_int(N),K2)),aa_int_int(modulo_modulo_int(N),K2)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P2,I4),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_4017_split__pos__lemma,axiom,
    ! [K2: int,P2: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P2,aa_int_int(divide_divide_int(N),K2)),aa_int_int(modulo_modulo_int(N),K2)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K2))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P2,I4),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_4018_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A4),B4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A4),N)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(B4),N)),zero_zero_int),one_one_int),zero_zero_int))),aa_int_int(divide_divide_int(B4),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_4019_even__flip__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),A)))
    <=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        <=> ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4020_even__flip__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M2),A)))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        <=> ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4021_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4022_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4023_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4024_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4025_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4026_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4027_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd),Xa2))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4028_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4029_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4030_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_4031_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4032_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_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4033_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4034_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4035_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4036_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4037_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_4038_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_4039_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_4040_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_4041_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4042_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_4043_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_4044_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_4045_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_4046_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_4047_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_4048_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_4049_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_4050_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_4051_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_4052_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_4053_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_4054_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_4055_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_4056_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_4057_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_4058_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_4059_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4060_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_4061_add__diff__cancel__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_4062_add__diff__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_4063_add__diff__cancel__right,axiom,
    ! [A: real,C2: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_4064_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_4065_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_4066_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_4067_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4068_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_4069_add__diff__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_4070_add__diff__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_4071_add__diff__cancel__left,axiom,
    ! [C2: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_4072_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_4073_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_4074_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_4075_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_4076_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_4077_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_4078_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_4079_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ).

% diff_Suc_Suc
tff(fact_4080_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),K2) ).

% Suc_diff_diff
tff(fact_4081_diff__self__eq__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),M2) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_4082_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_4083_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_4084_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ).

% diff_diff_left
tff(fact_4085_of__bool__less__eq__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P2)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( pp(P2)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4086_of__bool__less__eq__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)))
    <=> ( pp(P2)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4087_of__bool__less__eq__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( pp(P2)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4088_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n1201886186963655149omplex(P2) = zero_zero_complex )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_4089_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n3304061248610475627l_real(P2) = zero_zero_real )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_4090_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n2052037380579107095ol_rat(P2) = zero_zero_rat )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_4091_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P2) = zero_zero_nat )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_4092_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = zero_zero_int )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_4093_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_4094_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4095_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4096_of__bool__eq_I1_J,axiom,
    aa_bool_nat(zero_n2687167440665602831ol_nat,fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4097_of__bool__eq_I1_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4098_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P2)),zero_n3304061248610475627l_real(Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4099_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P2)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4100_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4101_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4102_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real(fTrue) = one_one_real ).

% of_bool_eq(2)
tff(fact_4103_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex(fTrue) = one_one_complex ).

% of_bool_eq(2)
tff(fact_4104_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat(fTrue) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4105_of__bool__eq_I2_J,axiom,
    aa_bool_nat(zero_n2687167440665602831ol_nat,fTrue) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4106_of__bool__eq_I2_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fTrue) = one_one_int ).

% of_bool_eq(2)
tff(fact_4107_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n3304061248610475627l_real(P2) = one_one_real )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_4108_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n1201886186963655149omplex(P2) = one_one_complex )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_4109_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n2052037380579107095ol_rat(P2) = one_one_rat )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_4110_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P2) = one_one_nat )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_4111_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = one_one_int )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_4112_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_4113_of__bool__or__iff,axiom,
    ! [P2: bool,Q: bool] : aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fdisj(P2),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)) ).

% of_bool_or_iff
tff(fact_4114_of__bool__or__iff,axiom,
    ! [P2: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(P2),Q)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_or_iff
tff(fact_4115_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_4116_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_4117_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_4118_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_4119_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_4120_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_4121_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4122_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4123_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4124_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_4125_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4126_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4127_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4128_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_4129_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_4130_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_4131_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_4132_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_4133_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_4134_div__diff,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(divide_divide_int(A),C2)),aa_int_int(divide_divide_int(B),C2)) ) ) ) ).

% div_diff
tff(fact_4135_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real(P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_4136_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_4137_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_bool_nat(zero_n2687167440665602831ol_nat,P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_4138_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_4139_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% zero_less_diff
tff(fact_4140_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_4141_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% diff_is_0_eq
tff(fact_4142_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P2)),one_one_real))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_4143_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P2)),one_one_rat))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_4144_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)),one_one_nat))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_4145_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),one_one_int))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_4146_of__bool__not__iff,axiom,
    ! [P2: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fNot,P2)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex(P2)) ).

% of_bool_not_iff
tff(fact_4147_of__bool__not__iff,axiom,
    ! [P2: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fNot,P2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real(P2)) ).

% of_bool_not_iff
tff(fact_4148_of__bool__not__iff,axiom,
    ! [P2: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,P2)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat(P2)) ).

% of_bool_not_iff
tff(fact_4149_of__bool__not__iff,axiom,
    ! [P2: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,P2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_bool_int(zero_n2684676970156552555ol_int,P2)) ).

% of_bool_not_iff
tff(fact_4150_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_4151_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_4152_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_4153_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_4154_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_4155_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_4156_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4157_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_4158_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_4159_even__diff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ).

% even_diff
tff(fact_4160_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_4161_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_int_int(divide_divide_int(aa_bool_int(zero_n2684676970156552555ol_int,B)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_4162_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_4163_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ).

% even_diff_nat
tff(fact_4164_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4165_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4166_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_4167_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_4168_bits__1__div__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_4169_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_4170_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4171_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4172_of__bool__eq__iff,axiom,
    ! [P: bool,Q2: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P) = aa_bool_nat(zero_n2687167440665602831ol_nat,Q2) )
    <=> ( pp(P)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4173_of__bool__eq__iff,axiom,
    ! [P: bool,Q2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = aa_bool_int(zero_n2684676970156552555ol_int,Q2) )
    <=> ( pp(P)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4174_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: 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,C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4175_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4176_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4177_diff__right__commute,axiom,
    ! [A: nat,C2: 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),C2)),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)),C2) ).

% diff_right_commute
tff(fact_4178_diff__right__commute,axiom,
    ! [A: int,C2: 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),C2)),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)),C2) ).

% diff_right_commute
tff(fact_4179_diff__right__commute,axiom,
    ! [A: real,C2: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) ).

% diff_right_commute
tff(fact_4180_diff__right__commute,axiom,
    ! [A: rat,C2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C2)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) ).

% diff_right_commute
tff(fact_4181_diff__commute,axiom,
    ! [I2: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),K2)),J) ).

% diff_commute
tff(fact_4182_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fconj(P2),Q)) = aa_complex_complex(times_times_complex(zero_n1201886186963655149omplex(P2)),zero_n1201886186963655149omplex(Q)) ).

% of_bool_conj
tff(fact_4183_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fconj(P2),Q)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(P2)),zero_n3304061248610475627l_real(Q)) ).

% of_bool_conj
tff(fact_4184_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fconj(P2),Q)) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat(P2)),zero_n2052037380579107095ol_rat(Q)) ).

% of_bool_conj
tff(fact_4185_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(P2),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)) ).

% of_bool_conj
tff(fact_4186_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(P2),Q)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_conj
tff(fact_4187_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4188_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4189_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: 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,C2),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4190_diff__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2))) ) ).

% diff_right_mono
tff(fact_4191_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),C2))) ) ).

% diff_right_mono
tff(fact_4192_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2))) ) ).

% diff_right_mono
tff(fact_4193_diff__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B))) ) ).

% diff_left_mono
tff(fact_4194_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C2),A)),aa_rat_rat(minus_minus_rat(C2),B))) ) ).

% diff_left_mono
tff(fact_4195_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))) ) ).

% diff_left_mono
tff(fact_4196_diff__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_mono
tff(fact_4197_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_mono
tff(fact_4198_diff__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_mono
tff(fact_4199_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_4200_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_4201_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_4202_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_4203_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2))) ) ).

% diff_strict_right_mono
tff(fact_4204_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),C2))) ) ).

% diff_strict_right_mono
tff(fact_4205_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2))) ) ).

% diff_strict_right_mono
tff(fact_4206_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B))) ) ).

% diff_strict_left_mono
tff(fact_4207_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(C2),A)),aa_rat_rat(minus_minus_rat(C2),B))) ) ).

% diff_strict_left_mono
tff(fact_4208_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))) ) ).

% diff_strict_left_mono
tff(fact_4209_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D)) ) ) ).

% diff_eq_diff_less
tff(fact_4210_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),D)) ) ) ).

% diff_eq_diff_less
tff(fact_4211_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: 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,C2),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),D)) ) ) ).

% diff_eq_diff_less
tff(fact_4212_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_strict_mono
tff(fact_4213_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_strict_mono
tff(fact_4214_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_strict_mono
tff(fact_4215_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),C2) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)) ).

% left_diff_distrib
tff(fact_4216_left__diff__distrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% left_diff_distrib
tff(fact_4217_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(times_times_rat(B),C2)) ).

% left_diff_distrib
tff(fact_4218_left__diff__distrib,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% left_diff_distrib
tff(fact_4219_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C2)) ).

% right_diff_distrib
tff(fact_4220_right__diff__distrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ).

% right_diff_distrib
tff(fact_4221_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C2)) ).

% right_diff_distrib
tff(fact_4222_right__diff__distrib,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% right_diff_distrib
tff(fact_4223_left__diff__distrib_H,axiom,
    ! [B: complex,C2: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B),C2)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(B),A)),aa_complex_complex(times_times_complex(C2),A)) ).

% left_diff_distrib'
tff(fact_4224_left__diff__distrib_H,axiom,
    ! [B: real,C2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)) ).

% left_diff_distrib'
tff(fact_4225_left__diff__distrib_H,axiom,
    ! [B: rat,C2: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C2)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C2),A)) ).

% left_diff_distrib'
tff(fact_4226_left__diff__distrib_H,axiom,
    ! [B: nat,C2: 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),C2)),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,C2),A)) ).

% left_diff_distrib'
tff(fact_4227_left__diff__distrib_H,axiom,
    ! [B: int,C2: 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),C2)),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,C2),A)) ).

% left_diff_distrib'
tff(fact_4228_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C2)) ).

% right_diff_distrib'
tff(fact_4229_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) ).

% right_diff_distrib'
tff(fact_4230_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C2)) ).

% right_diff_distrib'
tff(fact_4231_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C2: 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),C2)) = 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),C2)) ).

% right_diff_distrib'
tff(fact_4232_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% right_diff_distrib'
tff(fact_4233_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% diff_diff_eq
tff(fact_4234_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% diff_diff_eq
tff(fact_4235_diff__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ).

% diff_diff_eq
tff(fact_4236_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ).

% diff_diff_eq
tff(fact_4237_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B) = A )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4238_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) = A )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_4239_add__implies__diff,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B) = A )
     => ( C2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_4240_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C2),B) = A )
     => ( C2 = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_4241_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4242_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4243_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4244_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_add_eq
tff(fact_4245_diff__add__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ).

% diff_add_eq
tff(fact_4246_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),B) ).

% diff_add_eq
tff(fact_4247_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_diff_eq2
tff(fact_4248_diff__diff__eq2,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ).

% diff_diff_eq2
tff(fact_4249_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),B) ).

% diff_diff_eq2
tff(fact_4250_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% add_diff_eq
tff(fact_4251_add__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) ).

% add_diff_eq
tff(fact_4252_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) ).

% add_diff_eq
tff(fact_4253_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4254_eq__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4255_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C2),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4256_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C2 )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) ) ) ).

% diff_eq_eq
tff(fact_4257_diff__eq__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C2 )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B) ) ) ).

% diff_eq_eq
tff(fact_4258_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C2 )
    <=> ( A = aa_rat_rat(plus_plus_rat(C2),B) ) ) ).

% diff_eq_eq
tff(fact_4259_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4260_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4261_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(minus_minus_rat(A4),B) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4262_add__diff__add,axiom,
    ! [A: int,C2: 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),C2)),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,C2),D)) ).

% add_diff_add
tff(fact_4263_add__diff__add,axiom,
    ! [A: real,C2: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D)) ).

% add_diff_add
tff(fact_4264_add__diff__add,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C2),D)) ).

% add_diff_add
tff(fact_4265_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C2)),aa_rat_rat(divide_divide_rat(B),C2)) ).

% diff_divide_distrib
tff(fact_4266_diff__divide__distrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) ).

% diff_divide_distrib
tff(fact_4267_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z))) ) ) ).

% dvd_diff
tff(fact_4268_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))) ) ) ).

% dvd_diff
tff(fact_4269_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_rat_rat(minus_minus_rat(Y),Z))) ) ) ).

% dvd_diff
tff(fact_4270_zero__induct__lemma,axiom,
    ! [P2: fun_nat_bool,K2: nat,I2: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P2,N3)) )
       => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),I2))) ) ) ).

% zero_induct_lemma
tff(fact_4271_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2) = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
tff(fact_4272_minus__nat_Odiff__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),zero_zero_nat) = M2 ).

% minus_nat.diff_0
tff(fact_4273_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K2)) ) ).

% less_imp_diff_less
tff(fact_4274_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M2))) ) ) ).

% diff_less_mono2
tff(fact_4275_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M2))) ) ).

% diff_le_mono2
tff(fact_4276_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),B)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_4277_diff__le__self,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),M2)) ).

% diff_le_self
tff(fact_4278_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_4279_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4280_le__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ) ).

% le_diff_iff
tff(fact_4281_eq__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2) )
        <=> ( M2 = N ) ) ) ) ).

% eq_diff_iff
tff(fact_4282_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)),N) = M2 ).

% diff_add_inverse2
tff(fact_4283_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)),N) = M2 ).

% diff_add_inverse
tff(fact_4284_diff__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ).

% diff_cancel2
tff(fact_4285_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ).

% Nat.diff_cancel
tff(fact_4286_diff__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)) ).

% diff_mult_distrib
tff(fact_4287_diff__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) ).

% diff_mult_distrib2
tff(fact_4288_max__diff__distrib__left,axiom,
    ! [X: 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,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_4289_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_4290_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(X),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_4291_dvd__diff__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ) ) ).

% dvd_diff_nat
tff(fact_4292_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real(P2))) ).

% zero_less_eq_of_bool
tff(fact_4293_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P2))) ).

% zero_less_eq_of_bool
tff(fact_4294_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_bool_nat(zero_n2687167440665602831ol_nat,P2))) ).

% zero_less_eq_of_bool
tff(fact_4295_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P2))) ).

% zero_less_eq_of_bool
tff(fact_4296_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P2)),one_one_real)) ).

% of_bool_less_eq_one
tff(fact_4297_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P2)),one_one_rat)) ).

% of_bool_less_eq_one
tff(fact_4298_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)),one_one_nat)) ).

% of_bool_less_eq_one
tff(fact_4299_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),one_one_int)) ).

% of_bool_less_eq_one
tff(fact_4300_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n1201886186963655149omplex(P) = one_one_complex ) )
      & ( ~ pp(P)
       => ( zero_n1201886186963655149omplex(P) = zero_zero_complex ) ) ) ).

% of_bool_def
tff(fact_4301_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n3304061248610475627l_real(P) = one_one_real ) )
      & ( ~ pp(P)
       => ( zero_n3304061248610475627l_real(P) = zero_zero_real ) ) ) ).

% of_bool_def
tff(fact_4302_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n2052037380579107095ol_rat(P) = one_one_rat ) )
      & ( ~ pp(P)
       => ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat ) ) ) ).

% of_bool_def
tff(fact_4303_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( aa_bool_nat(zero_n2687167440665602831ol_nat,P) = one_one_nat ) )
      & ( ~ pp(P)
       => ( aa_bool_nat(zero_n2687167440665602831ol_nat,P) = zero_zero_nat ) ) ) ).

% of_bool_def
tff(fact_4304_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P) = one_one_int ) )
      & ( ~ pp(P)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P) = zero_zero_int ) ) ) ).

% of_bool_def
tff(fact_4305_split__of__bool,axiom,
    ! [P2: fun_complex_bool,P: bool] :
      ( pp(aa_complex_bool(P2,zero_n1201886186963655149omplex(P)))
    <=> ( ( pp(P)
         => pp(aa_complex_bool(P2,one_one_complex)) )
        & ( ~ pp(P)
         => pp(aa_complex_bool(P2,zero_zero_complex)) ) ) ) ).

% split_of_bool
tff(fact_4306_split__of__bool,axiom,
    ! [P2: fun_real_bool,P: bool] :
      ( pp(aa_real_bool(P2,zero_n3304061248610475627l_real(P)))
    <=> ( ( pp(P)
         => pp(aa_real_bool(P2,one_one_real)) )
        & ( ~ pp(P)
         => pp(aa_real_bool(P2,zero_zero_real)) ) ) ) ).

% split_of_bool
tff(fact_4307_split__of__bool,axiom,
    ! [P2: fun_rat_bool,P: bool] :
      ( pp(aa_rat_bool(P2,zero_n2052037380579107095ol_rat(P)))
    <=> ( ( pp(P)
         => pp(aa_rat_bool(P2,one_one_rat)) )
        & ( ~ pp(P)
         => pp(aa_rat_bool(P2,zero_zero_rat)) ) ) ) ).

% split_of_bool
tff(fact_4308_split__of__bool,axiom,
    ! [P2: fun_nat_bool,P: bool] :
      ( pp(aa_nat_bool(P2,aa_bool_nat(zero_n2687167440665602831ol_nat,P)))
    <=> ( ( pp(P)
         => pp(aa_nat_bool(P2,one_one_nat)) )
        & ( ~ pp(P)
         => pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ) ).

% split_of_bool
tff(fact_4309_split__of__bool,axiom,
    ! [P2: fun_int_bool,P: bool] :
      ( pp(aa_int_bool(P2,aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> ( ( pp(P)
         => pp(aa_int_bool(P2,one_one_int)) )
        & ( ~ pp(P)
         => pp(aa_int_bool(P2,zero_zero_int)) ) ) ) ).

% split_of_bool
tff(fact_4310_split__of__bool__asm,axiom,
    ! [P2: fun_complex_bool,P: bool] :
      ( pp(aa_complex_bool(P2,zero_n1201886186963655149omplex(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_complex_bool(P2,one_one_complex)) )
          | ( ~ pp(P)
            & ~ pp(aa_complex_bool(P2,zero_zero_complex)) ) ) ) ).

% split_of_bool_asm
tff(fact_4311_split__of__bool__asm,axiom,
    ! [P2: fun_real_bool,P: bool] :
      ( pp(aa_real_bool(P2,zero_n3304061248610475627l_real(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_real_bool(P2,one_one_real)) )
          | ( ~ pp(P)
            & ~ pp(aa_real_bool(P2,zero_zero_real)) ) ) ) ).

% split_of_bool_asm
tff(fact_4312_split__of__bool__asm,axiom,
    ! [P2: fun_rat_bool,P: bool] :
      ( pp(aa_rat_bool(P2,zero_n2052037380579107095ol_rat(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_rat_bool(P2,one_one_rat)) )
          | ( ~ pp(P)
            & ~ pp(aa_rat_bool(P2,zero_zero_rat)) ) ) ) ).

% split_of_bool_asm
tff(fact_4313_split__of__bool__asm,axiom,
    ! [P2: fun_nat_bool,P: bool] :
      ( pp(aa_nat_bool(P2,aa_bool_nat(zero_n2687167440665602831ol_nat,P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_nat_bool(P2,one_one_nat)) )
          | ( ~ pp(P)
            & ~ pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ) ).

% split_of_bool_asm
tff(fact_4314_split__of__bool__asm,axiom,
    ! [P2: fun_int_bool,P: bool] :
      ( pp(aa_int_bool(P2,aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_int_bool(P2,one_one_int)) )
          | ( ~ pp(P)
            & ~ pp(aa_int_bool(P2,zero_zero_int)) ) ) ) ).

% split_of_bool_asm
tff(fact_4315_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_4316_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_4317_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_4318_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_4319_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_4320_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_4321_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4322_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4323_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),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,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4324_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),C2)),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)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4325_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4326_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,C2),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4327_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),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,C2),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4328_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4329_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A))) ) ).

% le_add_diff
tff(fact_4330_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4331_le__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2)) ) ).

% le_diff_eq
tff(fact_4332_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C2)) ) ).

% le_diff_eq
tff(fact_4333_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2)) ) ).

% le_diff_eq
tff(fact_4334_diff__le__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))) ) ).

% diff_le_eq
tff(fact_4335_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C2),B))) ) ).

% diff_le_eq
tff(fact_4336_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))) ) ).

% diff_le_eq
tff(fact_4337_add__le__add__imp__diff__le,axiom,
    ! [I2: real,K2: real,N: real,J: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K2)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K2)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4338_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K2: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K2)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K2)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(N),K2)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4339_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K2: nat,N: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4340_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K2: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K2)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4341_add__le__imp__le__diff,axiom,
    ! [I2: real,K2: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K2)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_4342_add__le__imp__le__diff,axiom,
    ! [I2: rat,K2: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),aa_rat_rat(minus_minus_rat(N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_4343_add__le__imp__le__diff,axiom,
    ! [I2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_4344_add__le__imp__le__diff,axiom,
    ! [I2: int,K2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_4345_less__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2)) ) ).

% less_diff_eq
tff(fact_4346_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C2)) ) ).

% less_diff_eq
tff(fact_4347_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2)) ) ).

% less_diff_eq
tff(fact_4348_diff__less__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))) ) ).

% diff_less_eq
tff(fact_4349_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C2),B))) ) ).

% diff_less_eq
tff(fact_4350_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))) ) ).

% diff_less_eq
tff(fact_4351_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4352_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4353_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4354_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(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_4355_square__diff__square__factored,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),aa_complex_complex(times_times_complex(Y),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_complex_complex(minus_minus_complex(X),Y)) ).

% square_diff_square_factored
tff(fact_4356_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).

% square_diff_square_factored
tff(fact_4357_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_rat_rat(minus_minus_rat(X),Y)) ).

% square_diff_square_factored
tff(fact_4358_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_4359_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C2: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),C2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E2)),D) )
    <=> ( C2 = 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_4360_eq__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4361_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C2 = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4362_eq__add__iff2,axiom,
    ! [A: int,E2: int,C2: 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)),C2) = 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) )
    <=> ( C2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4363_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C2: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E2)),C2) = 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)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_4364_eq__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_4365_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_4366_eq__add__iff1,axiom,
    ! [A: int,E2: int,C2: 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)),C2) = 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)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_4367_mult__diff__mult,axiom,
    ! [X: complex,Y: complex,A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Y)),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),aa_complex_complex(minus_minus_complex(Y),B))),aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),A)),B)) ).

% mult_diff_mult
tff(fact_4368_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A)),B)) ).

% mult_diff_mult
tff(fact_4369_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),A)),B)) ).

% mult_diff_mult
tff(fact_4370_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A)),B)) ).

% mult_diff_mult
tff(fact_4371_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4372_dvd__minus__mod,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_4373_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_nat_nat(suc,M2))) ).

% diff_less_Suc
tff(fact_4374_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ) ) ).

% Suc_diff_Suc
tff(fact_4375_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),M2)) ) ) ).

% diff_less
tff(fact_4376_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ) ) ).

% Suc_diff_le
tff(fact_4377_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C2))) ) ) ).

% diff_less_mono
tff(fact_4378_less__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ) ).

% less_diff_iff
tff(fact_4379_diff__add__0,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = zero_zero_nat ).

% diff_add_0
tff(fact_4380_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = M2 ) ) ).

% add_diff_inverse_nat
tff(fact_4381_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J)) ) ).

% less_diff_conv
tff(fact_4382_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2) = K2 )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4383_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4384_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4385_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_4386_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2))) ) ).

% le_diff_conv
tff(fact_4387_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_4388_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% dvd_minus_self
tff(fact_4389_less__eq__dvd__minus,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% less_eq_dvd_minus
tff(fact_4390_dvd__diffD1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),M2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N)) ) ) ) ).

% dvd_diffD1
tff(fact_4391_dvd__diffD,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),M2)) ) ) ) ).

% dvd_diffD
tff(fact_4392_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) ) ) ).

% mod_geq
tff(fact_4393_mod__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = M2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) ) ) ) ).

% mod_if
tff(fact_4394_le__mod__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) ) ) ).

% le_mod_geq
tff(fact_4395_nat__minus__add__max,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M2) ).

% nat_minus_add_max
tff(fact_4396_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4397_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4398_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4399_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C2)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4400_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C2)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4401_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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)),C2)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4402_less__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4403_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4404_less__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4405_less__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C2)),D)) ) ).

% less_add_iff1
tff(fact_4406_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C2)),D)) ) ).

% less_add_iff1
tff(fact_4407_less__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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)),C2)),D)) ) ).

% less_add_iff1
tff(fact_4408_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = 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_4409_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4410_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4411_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4412_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4413_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4414_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4415_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4416_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4417_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4418_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4419_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4420_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_4421_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ).

% square_diff_one_factored
tff(fact_4422_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_4423_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_4424_inf__period_I3_J,axiom,
    ! [D: complex,D5: complex,T: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),D5))
     => ! [X3: complex,K4: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X3),T)))
        <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X3),aa_complex_complex(times_times_complex(K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_4425_inf__period_I3_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D5))
     => ! [X3: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_4426_inf__period_I3_J,axiom,
    ! [D: rat,D5: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D5))
     => ! [X3: rat,K4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_4427_inf__period_I3_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X3: int,K4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(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,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_4428_inf__period_I4_J,axiom,
    ! [D: complex,D5: complex,T: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),D5))
     => ! [X3: complex,K4: complex] :
          ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(X3),T)))
        <=> ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X3),aa_complex_complex(times_times_complex(K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_4429_inf__period_I4_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D5))
     => ! [X3: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_4430_inf__period_I4_J,axiom,
    ! [D: rat,D5: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D5))
     => ! [X3: rat,K4: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_4431_inf__period_I4_J,axiom,
    ! [D: int,D5: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D5))
     => ! [X3: int,K4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(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,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_4432_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_4433_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_4434_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_4435_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_4436_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_4437_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_4438_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_4439_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_4440_diff__Suc__less,axiom,
    ! [N: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I2))),N)) ) ).

% diff_Suc_less
tff(fact_4441_nat__diff__split__asm,axiom,
    ! [P2: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
            & ~ pp(aa_nat_bool(P2,zero_zero_nat)) )
          | ? [D6: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
              & ~ pp(aa_nat_bool(P2,D6)) ) ) ) ).

% nat_diff_split_asm
tff(fact_4442_nat__diff__split,axiom,
    ! [P2: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
         => pp(aa_nat_bool(P2,zero_zero_nat)) )
        & ! [D6: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
           => pp(aa_nat_bool(P2,D6)) ) ) ) ).

% nat_diff_split
tff(fact_4443_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2))) ) ) ).

% less_diff_conv2
tff(fact_4444_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_4445_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( 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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = 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,I2),J)),U)),M2)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_4446_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_4447_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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,I2),J)),U)),M2)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_4448_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4449_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( 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,I2),J)),U)),M2) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_4450_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4451_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),zero_zero_nat))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).

% exp_div_exp_eq
tff(fact_4452_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] : aa_int_int(divide_divide_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),zero_zero_int))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).

% exp_div_exp_eq
tff(fact_4453_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_4454_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_4455_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_4456_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_4457_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_4458_power__diff,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( A != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_4459_power__diff,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% power_diff
tff(fact_4460_power__diff,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_diff
tff(fact_4461_power__diff,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_4462_power__diff,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( A != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).

% power_diff
tff(fact_4463_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4464_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4465_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N)) ) ) ) ).

% div_geq
tff(fact_4466_div__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N)) ) ) ) ).

% div_if
tff(fact_4467_add__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = N ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_4468_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_4469_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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,I2),J)),U)),M2)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_4470_mult__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = zero_zero_nat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_4471_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M2)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_4472_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4473_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_4474_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_4475_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4476_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4477_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_4478_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2)),K2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K2)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_4479_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = 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,N),M2))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4480_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) = 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,N),M2))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4481_power__eq__if,axiom,
    ! [M2: nat,P: complex] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P),M2) = one_one_complex ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P),M2) = aa_complex_complex(times_times_complex(P),aa_nat_complex(power_power_complex(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4482_power__eq__if,axiom,
    ! [M2: nat,P: real] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P),M2) = one_one_real ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P),M2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P),aa_nat_real(aa_real_fun_nat_real(power_power_real,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4483_power__eq__if,axiom,
    ! [M2: nat,P: rat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P),M2) = one_one_rat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P),M2) = aa_rat_rat(times_times_rat(P),aa_nat_rat(power_power_rat(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4484_power__eq__if,axiom,
    ! [M2: nat,P: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),M2) = one_one_nat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4485_power__eq__if,axiom,
    ! [M2: nat,P: int] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P),M2) = one_one_int ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P),M2) = aa_int_int(aa_int_fun_int_int(times_times_int,P),aa_nat_int(power_power_int(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4486_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% power_minus_mult
tff(fact_4487_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% power_minus_mult
tff(fact_4488_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_4489_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_4490_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_4491_diff__le__diff__pow,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),N)))) ) ).

% diff_le_diff_pow
tff(fact_4492_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_4493_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_4494_bits__induct,axiom,
    ! [P2: fun_nat_bool,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => pp(aa_nat_bool(P2,A3)) )
     => ( ! [A3: nat,B3: bool] :
            ( pp(aa_nat_bool(P2,A3))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3)))) ) )
       => pp(aa_nat_bool(P2,A)) ) ) ).

% bits_induct
tff(fact_4495_bits__induct,axiom,
    ! [P2: fun_int_bool,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(divide_divide_int(A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => pp(aa_int_bool(P2,A3)) )
     => ( ! [A3: int,B3: bool] :
            ( pp(aa_int_bool(P2,A3))
           => ( ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3)))) ) )
       => pp(aa_int_bool(P2,A)) ) ) ).

% bits_induct
tff(fact_4496_int__power__div__base,axiom,
    ! [M2: nat,K2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K2),M2)),K2) = aa_nat_int(power_power_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4497_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ).

% exp_mod_exp
tff(fact_4498_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) ).

% exp_mod_exp
tff(fact_4499_power2__diff,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4500_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4501_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4502_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4503_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4504_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4505_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4506_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4507_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% even_mask_div_iff'
tff(fact_4508_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% even_mask_div_iff'
tff(fact_4509_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_4510_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% even_mask_div_iff
tff(fact_4511_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% even_mask_div_iff
tff(fact_4512_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2),R2) ) ) ) ).

% divmod_step_eq
tff(fact_4513_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_4514_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(T)),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,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_4515_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_4516_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_4517_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(X,Y) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Y)) ) ).

% diff_shunt_var
tff(fact_4518_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),Y)) ) ).

% diff_shunt_var
tff(fact_4519_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y)) ) ).

% diff_shunt_var
tff(fact_4520_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% diff_shunt_var
tff(fact_4521_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),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,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_4522_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_4523_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_4524_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_4525_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_4526_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4527_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_4528_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_4529_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_4530_Diff__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A4,bot_bo2099793752762293965at_nat) = A4 ).

% Diff_empty
tff(fact_4531_Diff__empty,axiom,
    ! [A4: set_real] : minus_minus_set_real(A4,bot_bot_set_real) = A4 ).

% Diff_empty
tff(fact_4532_Diff__empty,axiom,
    ! [A4: set_int] : minus_minus_set_int(A4,bot_bot_set_int) = A4 ).

% Diff_empty
tff(fact_4533_Diff__empty,axiom,
    ! [A4: set_nat] : minus_minus_set_nat(A4,bot_bot_set_nat) = A4 ).

% Diff_empty
tff(fact_4534_empty__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(bot_bo2099793752762293965at_nat,A4) = bot_bo2099793752762293965at_nat ).

% empty_Diff
tff(fact_4535_empty__Diff,axiom,
    ! [A4: set_real] : minus_minus_set_real(bot_bot_set_real,A4) = bot_bot_set_real ).

% empty_Diff
tff(fact_4536_empty__Diff,axiom,
    ! [A4: set_int] : minus_minus_set_int(bot_bot_set_int,A4) = bot_bot_set_int ).

% empty_Diff
tff(fact_4537_empty__Diff,axiom,
    ! [A4: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A4) = bot_bot_set_nat ).

% empty_Diff
tff(fact_4538_Diff__cancel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A4,A4) = bot_bo2099793752762293965at_nat ).

% Diff_cancel
tff(fact_4539_Diff__cancel,axiom,
    ! [A4: set_real] : minus_minus_set_real(A4,A4) = bot_bot_set_real ).

% Diff_cancel
tff(fact_4540_Diff__cancel,axiom,
    ! [A4: set_int] : minus_minus_set_int(A4,A4) = bot_bot_set_int ).

% Diff_cancel
tff(fact_4541_Diff__cancel,axiom,
    ! [A4: set_nat] : minus_minus_set_nat(A4,A4) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_4542_finite__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A4,B4))) ) ).

% finite_Diff
tff(fact_4543_finite__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4))) ) ).

% finite_Diff
tff(fact_4544_finite__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A4,B4))) ) ).

% finite_Diff
tff(fact_4545_finite__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A4,B4))) ) ).

% finite_Diff
tff(fact_4546_finite__Diff2,axiom,
    ! [B4: set_int,A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A4,B4)))
      <=> pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ).

% finite_Diff2
tff(fact_4547_finite__Diff2,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4)))
      <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ).

% finite_Diff2
tff(fact_4548_finite__Diff2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A4,B4)))
      <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4)) ) ) ).

% finite_Diff2
tff(fact_4549_finite__Diff2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A4,B4)))
      <=> pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ).

% finite_Diff2
tff(fact_4550_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),uminus1532241313380277803et_int(Y)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_4551_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% neg_le_iff_le
tff(fact_4552_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% neg_le_iff_le
tff(fact_4553_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% neg_le_iff_le
tff(fact_4554_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4555_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4556_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4557_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4558_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_4559_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_4560_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_4561_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4562_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_4563_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_4564_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_4565_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_4566_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4567_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4568_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4569_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4570_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4571_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4572_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% neg_less_iff_less
tff(fact_4573_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% neg_less_iff_less
tff(fact_4574_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_4575_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_4576_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_4577_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_4578_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_4579_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_4580_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_4581_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_4582_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_4583_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_4584_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_4585_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_4586_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_4587_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_4588_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_4589_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_4590_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_4591_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_4592_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_4593_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_4594_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_4595_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_4596_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_4597_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_4598_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_4599_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4600_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4601_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,X)),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% minus_dvd_iff
tff(fact_4602_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4603_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4604_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_rat_rat(uminus_uminus_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% dvd_minus_iff
tff(fact_4605_Diff__eq__empty__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat(A4,B4) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4)) ) ).

% Diff_eq_empty_iff
tff(fact_4606_Diff__eq__empty__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( minus_minus_set_real(A4,B4) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4)) ) ).

% Diff_eq_empty_iff
tff(fact_4607_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat(A4,B4) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4)) ) ).

% Diff_eq_empty_iff
tff(fact_4608_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A4,B4) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4)) ) ).

% Diff_eq_empty_iff
tff(fact_4609_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
          & ~ pp(aa_set_int_bool(ord_less_eq_set_int(L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4610_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
          & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4611_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
          & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4612_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 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
          & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4613_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
          & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4614_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 ) )
        | ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
          & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4615_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I2),set_or4548717258645045905et_nat(L,U)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),I2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4616_atLeastAtMost__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( member_set_int(I2,set_or370866239135849197et_int(L,U))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(L),I2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4617_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I2),set_or633870826150836451st_rat(L,U)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4618_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I2),set_or7049704709247886629st_num(L,U)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4619_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4620_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4621_atLeastAtMost__iff,axiom,
    ! [I2: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I2),set_or1222579329274155063t_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4622_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(ln_ln_real,Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_4623_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_4624_take__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_4625_take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_4626_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U))) ).

% finite_atLeastAtMost
tff(fact_4627_semiring__norm_I80_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M2)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% semiring_norm(80)
tff(fact_4628_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_4629_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_4630_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_4631_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_le_0_iff_le
tff(fact_4632_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_le_0_iff_le
tff(fact_4633_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_le_0_iff_le
tff(fact_4634_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_4635_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_4636_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_4637_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4638_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4639_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4640_less__neg__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_4641_less__neg__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_4642_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% less_neg_neg
tff(fact_4643_neg__less__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_pos
tff(fact_4644_neg__less__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_pos
tff(fact_4645_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_pos
tff(fact_4646_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_4647_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_4648_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_4649_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_0_iff_less
tff(fact_4650_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_0_iff_less
tff(fact_4651_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_0_iff_less
tff(fact_4652_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_4653_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_4654_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_4655_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4656_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_4657_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_4658_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_4659_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4660_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_4661_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_4662_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_4663_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_4664_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4665_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_4666_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4667_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_4668_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4669_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4670_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4671_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_4672_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_4673_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_4674_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_4675_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_4676_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_4677_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_4678_divide__minus1,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_4679_divide__minus1,axiom,
    ! [X: rat] : aa_rat_rat(divide_divide_rat(X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ).

% divide_minus1
tff(fact_4680_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4681_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4682_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4683_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4684_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4685_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4686_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4687_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4688_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4689_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4690_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4691_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4692_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C2,D))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        | ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),A))
          & pp(aa_set_int_bool(ord_less_eq_set_int(B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4693_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C2,D)))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4694_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C2,D)))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4695_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(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,C2),D)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4696_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C2),D)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4697_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C2,D)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4698_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_4699_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4700_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4701_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4702_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4703_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% infinite_Icc_iff
tff(fact_4704_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% infinite_Icc_iff
tff(fact_4705_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_4706_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_4707_take__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_4708_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_4709_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_4710_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_4711_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4712_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% zle_diff1_eq
tff(fact_4713_semiring__norm_I81_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M2)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% semiring_norm(81)
tff(fact_4714_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit1,N))) ).

% semiring_norm(77)
tff(fact_4715_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_4716_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_4717_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_4718_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4719_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_4720_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_4721_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_4722_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4723_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_4724_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_4725_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_4726_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4727_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_4728_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4729_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4730_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4731_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4732_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4733_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4734_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(2)
tff(fact_4735_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(2)
tff(fact_4736_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(2)
tff(fact_4737_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4738_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),one_one_int) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4739_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_4740_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_4741_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_4742_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_4743_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_4744_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M2)) ) ).

% neg_numeral_le_iff
tff(fact_4745_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M2)) ) ).

% neg_numeral_le_iff
tff(fact_4746_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M2)) ) ).

% neg_numeral_le_iff
tff(fact_4747_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M2)) ) ).

% neg_numeral_less_iff
tff(fact_4748_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M2)) ) ).

% neg_numeral_less_iff
tff(fact_4749_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M2)) ) ).

% neg_numeral_less_iff
tff(fact_4750_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_4751_semiring__norm_I74_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M2)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% semiring_norm(74)
tff(fact_4752_semiring__norm_I79_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M2)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% semiring_norm(79)
tff(fact_4753_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4754_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4755_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4756_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4757_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4758_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4759_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4760_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))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4761_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = B ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4762_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4763_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 )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4764_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = A )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4765_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4766_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4767_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4768_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat)))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4769_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4770_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4771_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4772_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4773_take__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4774_take__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),one_one_int) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4775_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4776_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4777_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4778_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4779_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_4780_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_4781_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: 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,M2)))),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))),M2)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_4782_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_4783_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: 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,M2)))),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))),M2)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4784_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_4785_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4786_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4787_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4788_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% take_bit_of_exp
tff(fact_4789_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_of_exp
tff(fact_4790_take__bit__of__2,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4791_take__bit__of__2,axiom,
    ! [N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4792_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_4793_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),X))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X)),Y)) ) ).

% compl_le_swap2
tff(fact_4794_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(X)))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X),uminus1532241313380277803et_int(Y))) ) ).

% compl_le_swap1
tff(fact_4795_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X))) ) ).

% compl_mono
tff(fact_4796_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_4797_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_4798_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_4799_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_4800_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_4801_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_4802_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_4803_take__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_4804_take__bit__add,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),B))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_4805_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),B) ) ) ) ).

% take_bit_tightened
tff(fact_4806_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M2: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),B) ) ) ) ).

% take_bit_tightened
tff(fact_4807_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2)),M2)) ).

% take_bit_nat_less_eq_self
tff(fact_4808_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4809_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% le_minus_iff
tff(fact_4810_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_minus_iff
tff(fact_4811_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% le_minus_iff
tff(fact_4812_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_le_iff
tff(fact_4813_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_le_iff
tff(fact_4814_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_le_iff
tff(fact_4815_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% le_imp_neg_le
tff(fact_4816_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_imp_neg_le
tff(fact_4817_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% le_imp_neg_le
tff(fact_4818_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% less_minus_iff
tff(fact_4819_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% less_minus_iff
tff(fact_4820_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% less_minus_iff
tff(fact_4821_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_less_iff
tff(fact_4822_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_less_iff
tff(fact_4823_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_less_iff
tff(fact_4824_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4825_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4826_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4827_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_4828_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_4829_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_4830_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_4831_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_4832_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_4833_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_4834_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_4835_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_4836_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_4837_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_4838_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(uminus_uminus_int,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4839_group__cancel_Oneg1,axiom,
    ! [A4: real,K2: real,A: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(uminus_uminus_real,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K2)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4840_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_4841_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_4842_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_4843_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_4844_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_4845_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_4846_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_4847_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_4848_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_4849_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).

% minus_divide_divide
tff(fact_4850_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ).

% minus_divide_divide
tff(fact_4851_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_4852_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_4853_Diff__infinite__finite,axiom,
    ! [T2: set_int,S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_4854_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_4855_Diff__infinite__finite,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2))
     => ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_4856_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,T2))) ) ) ).

% Diff_infinite_finite
tff(fact_4857_minus__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),zero_zero_int) = K2 ).

% minus_int_code(1)
tff(fact_4858_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_4859_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))),aa_real_real(uminus_uminus_real,X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_4860_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_4861_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_4862_psubset__imp__ex__mem,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A4),B4))
     => ? [B3: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),minus_1356011639430497352at_nat(B4,A4))) ) ).

% psubset_imp_ex_mem
tff(fact_4863_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B4: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A4),B4))
     => ? [B3: real] : pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4))) ) ).

% psubset_imp_ex_mem
tff(fact_4864_psubset__imp__ex__mem,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A4),B4))
     => ? [B3: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(B3),minus_2163939370556025621et_nat(B4,A4))) ) ).

% psubset_imp_ex_mem
tff(fact_4865_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),B4))
     => ? [B3: int] : pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A4))) ) ).

% psubset_imp_ex_mem
tff(fact_4866_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A4),B4))
     => ? [B3: nat] : pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B4,A4))) ) ).

% psubset_imp_ex_mem
tff(fact_4867_zdvd__zdiffD,axiom,
    ! [K2: int,M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,M2),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),M2)) ) ) ).

% zdvd_zdiffD
tff(fact_4868_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_4869_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_4870_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_4871_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_4872_take__bit__minus__small__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(uminus_uminus_int,K2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_4873_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B))) ) ).

% infinite_Icc
tff(fact_4874_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B))) ) ).

% infinite_Icc
tff(fact_4875_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_less_self
tff(fact_4876_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,N: nat,K2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),K2)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4877_not__take__bit__negative,axiom,
    ! [N: nat,K2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_4878_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_4879_signed__take__bit__take__bit,axiom,
    ! [M2: nat,N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,M2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) = aa_int_int(if_fun_int_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2),aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_nat_fun_int_int(bit_ri631733984087533419it_int,M2)),A) ).

% signed_take_bit_take_bit
tff(fact_4880_ex__nat__less,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
          & pp(aa_nat_bool(P2,M3)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)))
          & pp(aa_nat_bool(P2,X5)) ) ) ).

% ex_nat_less
tff(fact_4881_all__nat__less,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
         => pp(aa_nat_bool(P2,M3)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)))
         => pp(aa_nat_bool(P2,X5)) ) ) ).

% all_nat_less
tff(fact_4882_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4883_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4884_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4885_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_le_numeral
tff(fact_4886_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_le_numeral
tff(fact_4887_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_le_numeral
tff(fact_4888_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_4889_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_4890_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_4891_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% zero_neq_neg_numeral
tff(fact_4892_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% zero_neq_neg_numeral
tff(fact_4893_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4894_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4895_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4896_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_less_numeral
tff(fact_4897_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_less_numeral
tff(fact_4898_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_less_numeral
tff(fact_4899_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_4900_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_4901_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_4902_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_4903_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_4904_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_4905_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_4906_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_4907_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_4908_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_4909_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_4910_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_4911_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_4912_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_4913_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_4914_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_4915_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_4916_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_4917_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_4918_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_4919_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_4920_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_4921_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_4922_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_4923_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4924_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4925_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_4926_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_4927_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4928_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4929_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_4930_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_4931_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_4932_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_4933_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_4934_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_4935_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_4936_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4937_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4938_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_4939_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4940_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4941_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_4942_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_4943_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_4944_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_4945_group__cancel_Osub2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4946_group__cancel_Osub2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4947_group__cancel_Osub2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K2),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4948_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_4949_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_4950_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_4951_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_4952_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4953_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4954_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M2: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_4955_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M2: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_4956_take__bit__set__bit__eq,axiom,
    ! [N: nat,M2: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_4957_take__bit__set__bit__eq,axiom,
    ! [N: nat,M2: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_4958_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M2: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_4959_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M2: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_4960_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_4961_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_4962_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_4963_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_4964_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_4965_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_4966_subset__Compl__self__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(A4)))
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_4967_subset__Compl__self__eq,axiom,
    ! [A4: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),uminus612125837232591019t_real(A4)))
    <=> ( A4 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_4968_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(A4)))
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_4969_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(A4)))
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_4970_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_less_0_iff
tff(fact_4971_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_4972_int__le__induct,axiom,
    ! [I2: int,K2: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K2))
     => ( pp(aa_int_bool(P2,K2))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K2))
             => ( pp(aa_int_bool(P2,I3))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P2,I2)) ) ) ) ).

% int_le_induct
tff(fact_4973_int__less__induct,axiom,
    ! [I2: int,K2: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K2))
     => ( pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I3),K2))
             => ( pp(aa_int_bool(P2,I3))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P2,I2)) ) ) ) ).

% int_less_induct
tff(fact_4974_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
     => ( ( M2 = one_one_int )
        | ( M2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_4975_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
    <=> ( ( ( M2 = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M2 = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_4976_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_bound
tff(fact_4977_take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_4978_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_4979_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_4980_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_gt_zero
tff(fact_4981_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_4982_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C2,D))
    <=> ( ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
          | ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),A))
            & pp(aa_set_int_bool(ord_less_eq_set_int(B),D))
            & ( pp(aa_set_int_bool(ord_less_set_int(C2),A))
              | pp(aa_set_int_bool(ord_less_set_int(B),D)) ) ) )
        & pp(aa_set_int_bool(ord_less_eq_set_int(C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4983_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C2,D)))
    <=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
          | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D))
            & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A))
              | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),D)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4984_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C2,D)))
    <=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
          | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D))
            & ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A))
              | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),D)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4985_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(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,C2),D)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),D)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4986_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C2),D)))
    <=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
          | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),D)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4987_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C2,D)))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4988_take__bit__signed__take__bit,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_4989_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_4990_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_4991_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_4992_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_le_neg_numeral
tff(fact_4993_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_le_neg_numeral
tff(fact_4994_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_le_neg_numeral
tff(fact_4995_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_4996_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_4997_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_4998_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_less_neg_numeral
tff(fact_4999_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5000_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5001_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_5002_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_5003_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_5004_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_5005_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_5006_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_5007_numeral__Bit1,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_Bit1
tff(fact_5008_numeral__Bit1,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_Bit1
tff(fact_5009_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_5010_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_5011_numeral__Bit1,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_5012_numeral__Bit1,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_Bit1
tff(fact_5013_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_5014_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_5015_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_5016_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_5017_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_5018_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_5019_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)))) ).

% not_one_le_neg_numeral
tff(fact_5020_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)))) ).

% not_one_le_neg_numeral
tff(fact_5021_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)))) ).

% not_one_le_neg_numeral
tff(fact_5022_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_5023_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_5024_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_5025_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_5026_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_5027_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_5028_neg__one__le__numeral,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M2))) ).

% neg_one_le_numeral
tff(fact_5029_neg__one__le__numeral,axiom,
    ! [M2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M2))) ).

% neg_one_le_numeral
tff(fact_5030_neg__one__le__numeral,axiom,
    ! [M2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M2))) ).

% neg_one_le_numeral
tff(fact_5031_neg__numeral__le__one,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_5032_neg__numeral__le__one,axiom,
    ! [M2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_5033_neg__numeral__le__one,axiom,
    ! [M2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_5034_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5035_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5036_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5037_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)))) ).

% not_one_less_neg_numeral
tff(fact_5038_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)))) ).

% not_one_less_neg_numeral
tff(fact_5039_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)))) ).

% not_one_less_neg_numeral
tff(fact_5040_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_5041_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_5042_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_5043_neg__one__less__numeral,axiom,
    ! [M2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M2))) ).

% neg_one_less_numeral
tff(fact_5044_neg__one__less__numeral,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M2))) ).

% neg_one_less_numeral
tff(fact_5045_neg__one__less__numeral,axiom,
    ! [M2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M2))) ).

% neg_one_less_numeral
tff(fact_5046_neg__numeral__less__one,axiom,
    ! [M2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_5047_neg__numeral__less__one,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_5048_neg__numeral__less__one,axiom,
    ! [M2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_5049_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C2),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5050_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5051_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C2 = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C2),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5052_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C2: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C2 )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5053_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C2 )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5054_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C2 )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5055_minus__divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C2)) = A )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5056_minus__divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) = A )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5057_minus__divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)) = A )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5058_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C2)) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5059_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(uminus_uminus_real,B) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5060_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2)) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C2) = aa_rat_rat(uminus_uminus_rat,B) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5061_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_5062_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5063_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5064_take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_5065_take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_5066_plusinfinity,axiom,
    ! [D: int,P3: fun_int_bool,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K: int] :
            ( pp(aa_int_bool(P3,X4))
          <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
             => ( pp(aa_int_bool(P2,X4))
              <=> pp(aa_int_bool(P3,X4)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P3,X_1))
           => ? [X_12: int] : pp(aa_int_bool(P2,X_12)) ) ) ) ) ).

% plusinfinity
tff(fact_5067_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K: int] :
            ( pp(aa_int_bool(P1,X4))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
             => ( pp(aa_int_bool(P2,X4))
              <=> pp(aa_int_bool(P1,X4)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P1,X_1))
           => ? [X_12: int] : pp(aa_int_bool(P2,X_12)) ) ) ) ) ).

% minusinfinity
tff(fact_5068_int__induct,axiom,
    ! [P2: fun_int_bool,K2: int,I2: int] :
      ( pp(aa_int_bool(P2,K2))
     => ( ! [I3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),I3))
           => ( pp(aa_int_bool(P2,I3))
             => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int))) ) )
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K2))
             => ( pp(aa_int_bool(P2,I3))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P2,I2)) ) ) ) ).

% int_induct
tff(fact_5069_numeral__code_I3_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_code(3)
tff(fact_5070_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_code(3)
tff(fact_5071_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_code(3)
tff(fact_5072_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_code(3)
tff(fact_5073_numeral__code_I3_J,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat) ).

% numeral_code(3)
tff(fact_5074_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_code(3)
tff(fact_5075_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_5076_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_5077_take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_5078_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_5079_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_5080_minus__divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_5081_minus__divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_5082_neg__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5083_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5084_neg__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5085_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5086_pos__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5087_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5088_pos__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5089_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5090_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2) = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5091_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2) = B ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5092_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(B),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2) = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5093_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5094_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5095_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C2) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5096_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5097_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5098_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5099_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5100_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5101_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),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_5102_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = B ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_5103_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_5104_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5105_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5106_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5107_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),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_5108_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_5109_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_5110_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),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_5111_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_5112_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_5113_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_5114_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_5115_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5116_decr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_bool,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P2,X4))
           => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
         => ! [X3: int] :
              ( pp(aa_int_bool(P2,X3))
             => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_5117_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K2))
       => ( aa_int_int(modulo_modulo_int(K2),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_5118_verit__less__mono__div__int2,axiom,
    ! [A4: int,B4: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A4),B4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(B4),N)),aa_int_int(divide_divide_int(A4),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_5119_div__eq__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_5120_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2) = M2 ) ) ).

% take_bit_nat_eq_self
tff(fact_5121_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_5122_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2) = M2 )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_5123_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_5124_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_5125_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2)),K2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K2)) ) ).

% take_bit_int_less_self_iff
tff(fact_5126_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_5127_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_5128_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_5129_minus__divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_5130_minus__divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_5131_neg__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5132_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5133_neg__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5134_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5135_pos__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5136_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5137_pos__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5138_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C2))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5139_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5140_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5141_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5142_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5143_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5144_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5145_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5146_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5147_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M2)))),N) = 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))),M2)),N) ).

% Suc_div_eq_add3_div
tff(fact_5148_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5149_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( 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,N),K2)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5150_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5151_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5152_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M2)))),N) = 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))),M2)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_5153_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5154_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2) = K2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5155_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_5156_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_5157_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2)),M2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M2)) ) ).

% take_bit_nat_less_self_iff
tff(fact_5158_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2) = K2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_5159_take__bit__int__eq__self,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_5160_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5161_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5162_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5163_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5164_square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_5165_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_5166_square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_5167_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K2))
       => ( aa_int_int(divide_divide_int(K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_5168_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int))
       => ( aa_int_int(divide_divide_int(K2),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5169_take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K2))) ) ).

% take_bit_int_greater_eq
tff(fact_5170_take__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),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,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5171_take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5172_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5173_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5174_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = zero_zero_int ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5175_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_5176_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_5177_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5178_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5179_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5180_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5181_Bolzano,axiom,
    ! [A: real,B: real,P2: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [A3: real,B3: real,C: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),B3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,B3),C))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),B3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),C))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),C)) ) ) ) )
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
               => ? [D3: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
                    & ! [A3: real,B3: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),X4))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B3))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A3)),D3)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),B3)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A),B)) ) ) ) ).

% Bolzano
tff(fact_5182_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_er(L),Qr) ).

% divmod_step_def
tff(fact_5183_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_es(L)),Qr) ).

% divmod_step_def
tff(fact_5184_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U))) ).

% finite_atLeastAtMost_int
tff(fact_5185_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B)) ) ).

% case_prod_conv
tff(fact_5186_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(F,A),B) ).

% case_prod_conv
tff(fact_5187_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_5188_case__prod__conv,axiom,
    ! [F: fun_na4786964484885125043t_bool,A: nat,B: nat] : aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,A),B) ).

% case_prod_conv
tff(fact_5189_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_bool,A: int,B: int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),product_Pair_int_int(A,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(F,A),B)) ) ).

% case_prod_conv
tff(fact_5190_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique5052692396658037445od_int(M2,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M2),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5191_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique5055182867167087721od_nat(M2,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M2)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_5192_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5193_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5194_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5195_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_5196_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit0,N)) = produc4245557441103728435nt_int(aTP_Lamm_et,unique5052692396658037445od_int(M2,N)) ).

% divmod_algorithm_code(5)
tff(fact_5197_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_eu),unique5055182867167087721od_nat(M2,N)) ).

% divmod_algorithm_code(5)
tff(fact_5198_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit0,N)) = produc4245557441103728435nt_int(aTP_Lamm_ev,unique5052692396658037445od_int(M2,N)) ).

% divmod_algorithm_code(6)
tff(fact_5199_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ew),unique5055182867167087721od_nat(M2,N)) ).

% divmod_algorithm_code(6)
tff(fact_5200_prod_Ocase__distrib,axiom,
    ! [H: fun_bool_bool,F: fun_nat_fun_nat_bool,Prod: product_prod_nat_nat] :
      ( pp(aa_bool_bool(H,aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),Prod)))
    <=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_ex(H),F)),Prod)) ) ).

% prod.case_distrib
tff(fact_5201_prod_Ocase__distrib,axiom,
    ! [H: fun_bool_nat,F: fun_nat_fun_nat_bool,Prod: product_prod_nat_nat] : aa_bool_nat(H,aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun8134142976681035223at_nat(aTP_Lamm_ey(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5202_prod_Ocase__distrib,axiom,
    ! [H: fun_nat_bool,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] :
      ( pp(aa_nat_bool(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)))
    <=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(aa_fun1421985582829848961t_bool(aTP_Lamm_ez(H),F)),Prod)) ) ).

% prod.case_distrib
tff(fact_5203_prod_Ocase__distrib,axiom,
    ! [H: fun_nat_nat,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat_nat(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_fa(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5204_prod_Ocase__distrib,axiom,
    ! [H: fun_bool_bool,F: fun_int_fun_int_bool,Prod: product_prod_int_int] :
      ( pp(aa_bool_bool(H,aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),Prod)))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_fb(H),F)),Prod)) ) ).

% prod.case_distrib
tff(fact_5205_prod_Ocase__distrib,axiom,
    ! [H: fun_bo6148171183121378627t_bool,F: fun_nat_fun_nat_bool,Prod: product_prod_nat_nat] : aa_boo3692867584632636148t_bool(H,aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),Prod)) = aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aa_fun3328218794057202080t_bool(aTP_Lamm_fc(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5206_prod_Ocase__distrib,axiom,
    ! [H: fun_na4827981024643048934t_bool,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat4401263874371387303t_bool(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aa_fun9209259830566968027t_bool(aTP_Lamm_fd(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5207_prod_Ocase__distrib,axiom,
    ! [H: fun_fu5641360819561228055l_bool,F: fun_na4786964484885125043t_bool,Prod: product_prod_nat_nat] :
      ( pp(aa_fun8663595511526100224l_bool(H,aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),Prod)))
    <=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(aa_fun7513403936370313208t_bool(aTP_Lamm_fe(H),F)),Prod)) ) ).

% prod.case_distrib
tff(fact_5208_prod_Ocase__distrib,axiom,
    ! [H: fun_fu4450724535874491238ol_nat,F: fun_na4786964484885125043t_bool,Prod: product_prod_nat_nat] : aa_fun3448868659104709799ol_nat(H,aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun3048062428331337179at_nat(aTP_Lamm_ff(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5209_prod_Ocase__distrib,axiom,
    ! [H: fun_bo3682073979402284333at_nat,F: fun_nat_fun_nat_bool,Prod: product_prod_nat_nat] : aa_boo5791090712870310366at_nat(H,aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),Prod)) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aa_fun8974016276556183434at_nat(aTP_Lamm_fg(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5210_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_bool,X1: nat,X2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(F,X1),X2)) ) ).

% old.prod.case
tff(fact_5211_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_nat,X1: nat,X2: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5212_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5213_old_Oprod_Ocase,axiom,
    ! [F: fun_na4786964484885125043t_bool,X1: nat,X2: nat] : aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X1),X2) ).

% old.prod.case
tff(fact_5214_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_bool,X1: int,X2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),product_Pair_int_int(X1,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(F,X1),X2)) ) ).

% old.prod.case
tff(fact_5215_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_nat_fun_nat_bool,G: fun_nat_fun_nat_bool,P: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) = Q2 )
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,X4),Y3))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(G,X4),Y3)) ) )
     => ( ( P = Q2 )
       => ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),P))
        <=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(G),Q2)) ) ) ) ).

% split_cong
tff(fact_5216_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat,P: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_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) ) )
     => ( ( P = Q2 )
       => ( aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),P) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(G),Q2) ) ) ) ).

% split_cong
tff(fact_5217_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_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) ) )
     => ( ( P = Q2 )
       => ( aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),P) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(G),Q2) ) ) ) ).

% split_cong
tff(fact_5218_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na4786964484885125043t_bool,G: fun_na4786964484885125043t_bool,P: product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) = Q2 )
         => ( aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X4),Y3) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(G,X4),Y3) ) )
     => ( ( P = Q2 )
       => ( aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),P) = aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(G),Q2) ) ) ) ).

% split_cong
tff(fact_5219_split__cong,axiom,
    ! [Q2: product_prod_int_int,F: fun_int_fun_int_bool,G: fun_int_fun_int_bool,P: product_prod_int_int] :
      ( ! [X4: int,Y3: int] :
          ( ( product_Pair_int_int(X4,Y3) = Q2 )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(F,X4),Y3))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(G,X4),Y3)) ) )
     => ( ( P = Q2 )
       => ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),P))
        <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(G),Q2)) ) ) ) ).

% split_cong
tff(fact_5220_case__prodE2,axiom,
    ! [Q: fun_bool_bool,P2: fun_nat_fun_nat_bool,Z: product_prod_nat_nat] :
      ( pp(aa_bool_bool(Q,aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(P2),Z)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_bool_bool(Q,aa_nat_bool(aa_nat_fun_nat_bool(P2,X4),Y3))) ) ) ).

% case_prodE2
tff(fact_5221_case__prodE2,axiom,
    ! [Q: fun_nat_bool,P2: fun_nat_fun_nat_nat,Z: product_prod_nat_nat] :
      ( pp(aa_nat_bool(Q,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(P2),Z)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_nat_bool(Q,aa_nat_nat(aa_nat_fun_nat_nat(P2,X4),Y3))) ) ) ).

% case_prodE2
tff(fact_5222_case__prodE2,axiom,
    ! [Q: fun_fu7919257609373989805t_bool,P2: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( pp(aa_fun4519828053337421206t_bool(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P2),Z)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_fun4519828053337421206t_bool(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P2,X4),Y3))) ) ) ).

% case_prodE2
tff(fact_5223_case__prodE2,axiom,
    ! [Q: fun_fu5641360819561228055l_bool,P2: fun_na4786964484885125043t_bool,Z: product_prod_nat_nat] :
      ( pp(aa_fun8663595511526100224l_bool(Q,aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(P2),Z)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_fun8663595511526100224l_bool(Q,aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(P2,X4),Y3))) ) ) ).

% case_prodE2
tff(fact_5224_case__prodE2,axiom,
    ! [Q: fun_bool_bool,P2: fun_int_fun_int_bool,Z: product_prod_int_int] :
      ( pp(aa_bool_bool(Q,aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(P2),Z)))
     => ~ ! [X4: int,Y3: int] :
            ( ( Z = product_Pair_int_int(X4,Y3) )
           => ~ pp(aa_bool_bool(Q,aa_int_bool(aa_int_fun_int_bool(P2,X4),Y3))) ) ) ).

% case_prodE2
tff(fact_5225_case__prod__eta,axiom,
    ! [F: fun_Pr7757163693017731481t_bool] : produc8955392012077580105t_bool(aTP_Lamm_fh(F)) = F ).

% case_prod_eta
tff(fact_5226_case__prod__eta,axiom,
    ! [F: fun_Pr2602117396471556196at_nat] : produc6842872674320459806at_nat(aTP_Lamm_fi(F)) = F ).

% case_prod_eta
tff(fact_5227_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_fj(F)) = F ).

% case_prod_eta
tff(fact_5228_case__prod__eta,axiom,
    ! [F: fun_Pr248683899324676269t_bool] : produc5023667602222594023t_bool(aTP_Lamm_fk(F)) = F ).

% case_prod_eta
tff(fact_5229_case__prod__eta,axiom,
    ! [F: fun_Pr803113274115125473t_bool] : produc8560705615207772305t_bool(aTP_Lamm_fl(F)) = F ).

% case_prod_eta
tff(fact_5230_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_bool,G: fun_Pr7757163693017731481t_bool] :
      ( ! [X4: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,X4),Y3))
        <=> pp(aa_Pro2232559814647311250t_bool(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))) )
     => ( produc8955392012077580105t_bool(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5231_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(product_Pair_nat_nat(X4),Y3))
     => ( produc6842872674320459806at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5232_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(product_Pair_nat_nat(X4),Y3))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5233_cond__case__prod__eta,axiom,
    ! [F: fun_na4786964484885125043t_bool,G: fun_Pr248683899324676269t_bool] :
      ( ! [X4: nat,Y3: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X4),Y3) = aa_Pro595088089689268830t_bool(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))
     => ( produc5023667602222594023t_bool(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5234_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_bool,G: fun_Pr803113274115125473t_bool] :
      ( ! [X4: int,Y3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(F,X4),Y3))
        <=> pp(aa_Pro4501881432599481050t_bool(G,product_Pair_int_int(X4,Y3))) )
     => ( produc8560705615207772305t_bool(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5235_periodic__finite__ex,axiom,
    ! [D: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K: int] :
            ( pp(aa_int_bool(P2,X4))
          <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
        <=> ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D)))
              & pp(aa_int_bool(P2,X5)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5236_aset_I7_J,axiom,
    ! [D5: int,A4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A4))
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5))) ) ) ) ).

% aset(7)
tff(fact_5237_aset_I5_J,axiom,
    ! [D5: int,T: int,A4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),T)) ) ) ) ) ).

% aset(5)
tff(fact_5238_aset_I4_J,axiom,
    ! [D5: int,T: int,A4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X3 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5) != T ) ) ) ) ) ).

% aset(4)
tff(fact_5239_aset_I3_J,axiom,
    ! [D5: int,T: int,A4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X3 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5) = T ) ) ) ) ) ).

% aset(3)
tff(fact_5240_bset_I7_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),B4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5))) ) ) ) ) ).

% bset(7)
tff(fact_5241_bset_I5_J,axiom,
    ! [D5: int,B4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B4))
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),T)) ) ) ) ).

% bset(5)
tff(fact_5242_bset_I4_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),B4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X3 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5) != T ) ) ) ) ) ).

% bset(4)
tff(fact_5243_bset_I3_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X3 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5) = T ) ) ) ) ) ).

% bset(3)
tff(fact_5244_bset_I6_J,axiom,
    ! [D5: int,B4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B4))
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5)),T)) ) ) ) ).

% bset(6)
tff(fact_5245_bset_I8_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D5))) ) ) ) ) ).

% bset(8)
tff(fact_5246_aset_I6_J,axiom,
    ! [D5: int,T: int,A4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A4))
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A4))
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5)),T)) ) ) ) ) ).

% aset(6)
tff(fact_5247_aset_I8_J,axiom,
    ! [D5: int,A4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A4))
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D5))) ) ) ) ).

% aset(8)
tff(fact_5248_cpmi,axiom,
    ! [D5: int,P2: fun_int_bool,P3: fun_int_bool,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(P2,X4))
            <=> pp(aa_int_bool(P3,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),B4))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P2,X4))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5))) ) )
         => ( ! [X4: int,K: int] :
                ( pp(aa_int_bool(P3,X4))
              <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D5)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
            <=> ( ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
                    & pp(aa_int_bool(P3,X5)) )
                | ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),B4))
                        & pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X5))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5249_cppi,axiom,
    ! [D5: int,P2: fun_int_bool,P3: fun_int_bool,A4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(P2,X4))
            <=> pp(aa_int_bool(P3,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),A4))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P2,X4))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5))) ) )
         => ( ! [X4: int,K: int] :
                ( pp(aa_int_bool(P3,X4))
              <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D5)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
            <=> ( ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
                    & pp(aa_int_bool(P3,X5)) )
                | ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D5)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),A4))
                        & pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X5))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5250_divmod__def,axiom,
    ! [M2: num,N: num] : unique5052692396658037445od_int(M2,N) = product_Pair_int_int(aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_5251_divmod__def,axiom,
    ! [M2: num,N: num] : unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod_def
tff(fact_5252_divmod_H__nat__def,axiom,
    ! [M2: num,N: num] : unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod'_nat_def
tff(fact_5253_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_es(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5254_divmod__divmod__step,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5052692396658037445od_int(M2,N) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M2)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5052692396658037445od_int(M2,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M2,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5255_divmod__divmod__step,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M2)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5055182867167087721od_nat(M2,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M2,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5256_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5257_divmod__nat__if,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) )
       => ( divmod_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M2) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) )
       => ( divmod_nat(M2,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fm),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_5258_of__int__code__if,axiom,
    ! [K2: int] :
      ( ( ( K2 = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K2) = zero_zero_complex ) )
      & ( ( K2 != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K2) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K2))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K2) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5259_of__int__code__if,axiom,
    ! [K2: int] :
      ( ( ( K2 = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K2) = zero_zero_int ) )
      & ( ( K2 != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K2) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K2))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K2) = aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5260_of__int__code__if,axiom,
    ! [K2: int] :
      ( ( ( K2 = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K2) = zero_zero_real ) )
      & ( ( K2 != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K2) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K2))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K2) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5261_of__int__code__if,axiom,
    ! [K2: int] :
      ( ( ( K2 = zero_zero_int )
       => ( aa_int_rat(ring_1_of_int_rat,K2) = zero_zero_rat ) )
      & ( ( K2 != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K2) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K2))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K2) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5262_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_5263_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_5264_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_5265_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5266_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5267_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5268_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_5269_of__int__eq__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,W2) = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( W2 = Z ) ) ).

% of_int_eq_iff
tff(fact_5270_case__prodI2,axiom,
    ! [P: produc859450856879609959at_nat,C2: fun_Pr248683899324676269t_bool] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( ( P = produc6161850002892822231at_nat(A3,B3) )
         => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(C2,A3),B3)) )
     => pp(aa_Pro6098352494050694700t_bool(produc6738493412065443883t_bool(C2),P)) ) ).

% case_prodI2
tff(fact_5271_case__prodI2,axiom,
    ! [P: produc1319942482725812455at_nat,C2: fun_se5010674451832123565t_bool] :
      ( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
          ( ( P = produc9060074326276436823at_nat(A3,B3) )
         => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(C2,A3),B3)) )
     => pp(aa_Pro4324259192382070444t_bool(produc3930200643136266155t_bool(C2),P)) ) ).

% case_prodI2
tff(fact_5272_case__prodI2,axiom,
    ! [P: produc3843707927480180839at_nat,C2: fun_se1534075524646452397t_bool] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( P = produc2922128104949294807at_nat(A3,B3) )
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(C2,A3),B3)) )
     => pp(aa_Pro1809015098595223084t_bool(produc4641332057840980011t_bool(C2),P)) ) ).

% case_prodI2
tff(fact_5273_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C2: fun_nat_fun_nat_bool] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(C2,A3),B3)) )
     => pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(C2),P)) ) ).

% case_prodI2
tff(fact_5274_case__prodI2,axiom,
    ! [P: product_prod_int_int,C2: fun_int_fun_int_bool] :
      ( ! [A3: int,B3: int] :
          ( ( P = product_Pair_int_int(A3,B3) )
         => pp(aa_int_bool(aa_int_fun_int_bool(C2,A3),B3)) )
     => pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(C2),P)) ) ).

% case_prodI2
tff(fact_5275_case__prodI,axiom,
    ! [F: fun_Pr248683899324676269t_bool,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(F,A),B))
     => pp(aa_Pro6098352494050694700t_bool(produc6738493412065443883t_bool(F),produc6161850002892822231at_nat(A,B))) ) ).

% case_prodI
tff(fact_5276_case__prodI,axiom,
    ! [F: fun_se5010674451832123565t_bool,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(F,A),B))
     => pp(aa_Pro4324259192382070444t_bool(produc3930200643136266155t_bool(F),produc9060074326276436823at_nat(A,B))) ) ).

% case_prodI
tff(fact_5277_case__prodI,axiom,
    ! [F: fun_se1534075524646452397t_bool,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(F,A),B))
     => pp(aa_Pro1809015098595223084t_bool(produc4641332057840980011t_bool(F),produc2922128104949294807at_nat(A,B))) ) ).

% case_prodI
tff(fact_5278_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B))
     => pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ) ).

% case_prodI
tff(fact_5279_case__prodI,axiom,
    ! [F: fun_int_fun_int_bool,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(F,A),B))
     => pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),product_Pair_int_int(A,B))) ) ).

% case_prodI
tff(fact_5280_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: real,C2: fun_na312559527389416956t_real] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => pp(aa_set_real_bool(member_real(Z),aa_nat_set_real(aa_nat3669977791431069885t_real(C2,A3),B3))) )
     => pp(aa_set_real_bool(member_real(Z),produc3668448655016342576t_real(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5281_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: nat,C2: fun_na6128946301132549280et_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => pp(aa_set_nat_bool(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C2,A3),B3))) )
     => pp(aa_set_nat_bool(member_nat(Z),produc6189476227299908564et_nat(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5282_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: int,C2: fun_na1315397882344630524et_int] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => pp(aa_set_int_bool(member_int(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C2,A3),B3))) )
     => pp(aa_set_int_bool(member_int(Z),produc2011625207790711856et_int(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5283_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: real,C2: fun_in6553099857819052212t_real] :
      ( ! [A3: int,B3: int] :
          ( ( P = product_Pair_int_int(A3,B3) )
         => pp(aa_set_real_bool(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C2,A3),B3))) )
     => pp(aa_set_real_bool(member_real(Z),produc6452406959799940328t_real(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5284_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: nat,C2: fun_in2630332840903238488et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P = product_Pair_int_int(A3,B3) )
         => pp(aa_set_nat_bool(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,A3),B3))) )
     => pp(aa_set_nat_bool(member_nat(Z),produc4251311855443802252et_nat(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5285_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: int,C2: fun_in7040156458970095540et_int] :
      ( ! [A3: int,B3: int] :
          ( ( P = product_Pair_int_int(A3,B3) )
         => pp(aa_set_int_bool(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C2,A3),B3))) )
     => pp(aa_set_int_bool(member_int(Z),produc73460835934605544et_int(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5286_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: set_nat,C2: fun_na2051478540833925974et_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => pp(aa_set_set_nat_bool(member_set_nat(Z),aa_nat_set_set_nat(aa_nat4995987175846560023et_nat(C2,A3),B3))) )
     => pp(aa_set_set_nat_bool(member_set_nat(Z),produc8404753619367356554et_nat(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5287_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: set_nat,C2: fun_in7207726656141197838et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P = product_Pair_int_int(A3,B3) )
         => pp(aa_set_set_nat_bool(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,A3),B3))) )
     => pp(aa_set_set_nat_bool(member_set_nat(Z),produc5233655623923918146et_nat(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5288_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: product_prod_nat_nat,C2: fun_na1790740735814032627at_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_nat7062706319057040595at_nat(aa_nat7495903657192774236at_nat(C2,A3),B3))) )
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc8197505143624133779at_nat(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5289_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: product_prod_nat_nat,C2: fun_in1904380326158164027at_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P = product_Pair_int_int(A3,B3) )
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C2,A3),B3))) )
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C2,P))) ) ).

% mem_case_prodI2
tff(fact_5290_mem__case__prodI,axiom,
    ! [Z: real,C2: fun_na312559527389416956t_real,A: nat,B: nat] :
      ( pp(aa_set_real_bool(member_real(Z),aa_nat_set_real(aa_nat3669977791431069885t_real(C2,A),B)))
     => pp(aa_set_real_bool(member_real(Z),produc3668448655016342576t_real(C2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ) ).

% mem_case_prodI
tff(fact_5291_mem__case__prodI,axiom,
    ! [Z: nat,C2: fun_na6128946301132549280et_nat,A: nat,B: nat] :
      ( pp(aa_set_nat_bool(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C2,A),B)))
     => pp(aa_set_nat_bool(member_nat(Z),produc6189476227299908564et_nat(C2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ) ).

% mem_case_prodI
tff(fact_5292_mem__case__prodI,axiom,
    ! [Z: int,C2: fun_na1315397882344630524et_int,A: nat,B: nat] :
      ( pp(aa_set_int_bool(member_int(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C2,A),B)))
     => pp(aa_set_int_bool(member_int(Z),produc2011625207790711856et_int(C2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ) ).

% mem_case_prodI
tff(fact_5293_mem__case__prodI,axiom,
    ! [Z: real,C2: fun_in6553099857819052212t_real,A: int,B: int] :
      ( pp(aa_set_real_bool(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C2,A),B)))
     => pp(aa_set_real_bool(member_real(Z),produc6452406959799940328t_real(C2,product_Pair_int_int(A,B)))) ) ).

% mem_case_prodI
tff(fact_5294_mem__case__prodI,axiom,
    ! [Z: nat,C2: fun_in2630332840903238488et_nat,A: int,B: int] :
      ( pp(aa_set_nat_bool(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,A),B)))
     => pp(aa_set_nat_bool(member_nat(Z),produc4251311855443802252et_nat(C2,product_Pair_int_int(A,B)))) ) ).

% mem_case_prodI
tff(fact_5295_mem__case__prodI,axiom,
    ! [Z: int,C2: fun_in7040156458970095540et_int,A: int,B: int] :
      ( pp(aa_set_int_bool(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C2,A),B)))
     => pp(aa_set_int_bool(member_int(Z),produc73460835934605544et_int(C2,product_Pair_int_int(A,B)))) ) ).

% mem_case_prodI
tff(fact_5296_mem__case__prodI,axiom,
    ! [Z: set_nat,C2: fun_na2051478540833925974et_nat,A: nat,B: nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(Z),aa_nat_set_set_nat(aa_nat4995987175846560023et_nat(C2,A),B)))
     => pp(aa_set_set_nat_bool(member_set_nat(Z),produc8404753619367356554et_nat(C2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ) ).

% mem_case_prodI
tff(fact_5297_mem__case__prodI,axiom,
    ! [Z: set_nat,C2: fun_in7207726656141197838et_nat,A: int,B: int] :
      ( pp(aa_set_set_nat_bool(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,A),B)))
     => pp(aa_set_set_nat_bool(member_set_nat(Z),produc5233655623923918146et_nat(C2,product_Pair_int_int(A,B)))) ) ).

% mem_case_prodI
tff(fact_5298_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C2: fun_na1790740735814032627at_nat,A: nat,B: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_nat7062706319057040595at_nat(aa_nat7495903657192774236at_nat(C2,A),B)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc8197505143624133779at_nat(C2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ) ).

% mem_case_prodI
tff(fact_5299_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C2: fun_in1904380326158164027at_nat,A: int,B: int] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C2,A),B)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C2,product_Pair_int_int(A,B)))) ) ).

% mem_case_prodI
tff(fact_5300_case__prodI2_H,axiom,
    ! [P: product_prod_nat_nat,C2: fun_na4786964484885125043t_bool,X: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) = P )
         => pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(C2,A3),B3),X)) )
     => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(C2),P),X)) ) ).

% case_prodI2'
tff(fact_5301_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_5302_of__int__of__bool,axiom,
    ! [P2: bool] : aa_int_real(ring_1_of_int_real,aa_bool_int(zero_n2684676970156552555ol_int,P2)) = zero_n3304061248610475627l_real(P2) ).

% of_int_of_bool
tff(fact_5303_of__int__of__bool,axiom,
    ! [P2: bool] : aa_int_rat(ring_1_of_int_rat,aa_bool_int(zero_n2684676970156552555ol_int,P2)) = zero_n2052037380579107095ol_rat(P2) ).

% of_int_of_bool
tff(fact_5304_of__int__of__bool,axiom,
    ! [P2: bool] : aa_int_int(ring_1_of_int_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)) = aa_bool_int(zero_n2684676970156552555ol_int,P2) ).

% of_int_of_bool
tff(fact_5305_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_5306_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_5307_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% tanh_real_less_iff
tff(fact_5308_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_5309_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5310_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5311_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5312_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_5313_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_5314_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_5315_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5316_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_5317_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_5318_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_5319_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5320_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5321_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5322_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% of_int_le_iff
tff(fact_5323_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5324_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5325_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5326_of__int__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K2)) = aa_num_rat(numeral_numeral_rat,K2) ).

% of_int_numeral
tff(fact_5327_of__int__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,K2) ).

% of_int_numeral
tff(fact_5328_of__int__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K2)) = aa_num_real(numeral_numeral_real,K2) ).

% of_int_numeral
tff(fact_5329_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5330_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5331_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% of_int_less_iff
tff(fact_5332_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_5333_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_5334_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_5335_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5336_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_5337_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_5338_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_5339_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_5340_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_5341_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_5342_of__int__mult,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_5343_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_5344_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_5345_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_5346_of__int__add,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_5347_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_5348_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_5349_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_5350_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_5351_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_5352_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_5353_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_5354_of__int__diff,axiom,
    ! [W2: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_5355_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_5356_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_5357_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_5358_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5359_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5360_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5361_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5362_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) = aa_int_rat(ring_1_of_int_rat,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5363_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5364_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5365_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5366_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).

% of_int_power
tff(fact_5367_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),N) ).

% of_int_power
tff(fact_5368_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),N)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),N) ).

% of_int_power
tff(fact_5369_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),N) ).

% of_int_power
tff(fact_5370_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% tanh_real_neg_iff
tff(fact_5371_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% tanh_real_pos_iff
tff(fact_5372_less__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K2))) ) ).

% less_Suc_numeral
tff(fact_5373_less__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K2)),N)) ) ).

% less_numeral_Suc
tff(fact_5374_le__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K2))) ) ).

% le_Suc_numeral
tff(fact_5375_le__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K2)),N)) ) ).

% le_numeral_Suc
tff(fact_5376_max__Suc__numeral,axiom,
    ! [N: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K2))) ).

% max_Suc_numeral
tff(fact_5377_max__numeral__Suc,axiom,
    ! [K2: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K2)),N)) ).

% max_numeral_Suc
tff(fact_5378_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5379_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5380_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5381_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5382_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5383_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5384_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5385_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5386_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5387_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5388_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5389_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5390_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5391_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5392_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5393_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5394_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5395_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5396_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5397_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5398_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5399_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5400_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5401_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5402_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5403_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5404_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5405_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5406_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5407_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5408_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5409_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5410_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5411_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5412_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5413_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5414_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5415_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5416_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5417_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5418_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5419_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5420_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5421_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5422_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5423_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5424_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5425_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5426_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5427_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5428_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5429_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5430_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5431_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5432_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5433_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5434_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5435_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5436_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5437_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5438_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5439_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5440_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5441_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5442_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5443_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5444_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5445_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5446_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5447_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5448_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5449_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5450_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5451_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5452_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5453_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5454_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5455_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5456_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5457_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5458_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5459_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5460_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5461_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5462_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5463_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_le_of_int
tff(fact_5464_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_le_of_int
tff(fact_5465_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_5466_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_less_of_int
tff(fact_5467_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_less_of_int
tff(fact_5468_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X)) ).

% ex_of_int_less
tff(fact_5469_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)) ).

% ex_of_int_less
tff(fact_5470_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,X)) ).

% mult_of_int_commute
tff(fact_5471_mult__of__int__commute,axiom,
    ! [X: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_5472_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X)) ).

% mult_of_int_commute
tff(fact_5473_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_5474_mem__case__prodE,axiom,
    ! [Z: real,C2: fun_na312559527389416956t_real,P: product_prod_nat_nat] :
      ( pp(aa_set_real_bool(member_real(Z),produc3668448655016342576t_real(C2,P)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_set_real_bool(member_real(Z),aa_nat_set_real(aa_nat3669977791431069885t_real(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5475_mem__case__prodE,axiom,
    ! [Z: nat,C2: fun_na6128946301132549280et_nat,P: product_prod_nat_nat] :
      ( pp(aa_set_nat_bool(member_nat(Z),produc6189476227299908564et_nat(C2,P)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_set_nat_bool(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5476_mem__case__prodE,axiom,
    ! [Z: int,C2: fun_na1315397882344630524et_int,P: product_prod_nat_nat] :
      ( pp(aa_set_int_bool(member_int(Z),produc2011625207790711856et_int(C2,P)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_set_int_bool(member_int(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5477_mem__case__prodE,axiom,
    ! [Z: real,C2: fun_in6553099857819052212t_real,P: product_prod_int_int] :
      ( pp(aa_set_real_bool(member_real(Z),produc6452406959799940328t_real(C2,P)))
     => ~ ! [X4: int,Y3: int] :
            ( ( P = product_Pair_int_int(X4,Y3) )
           => ~ pp(aa_set_real_bool(member_real(Z),aa_int_set_real(aa_int4847878404602763637t_real(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5478_mem__case__prodE,axiom,
    ! [Z: nat,C2: fun_in2630332840903238488et_nat,P: product_prod_int_int] :
      ( pp(aa_set_nat_bool(member_nat(Z),produc4251311855443802252et_nat(C2,P)))
     => ~ ! [X4: int,Y3: int] :
            ( ( P = product_Pair_int_int(X4,Y3) )
           => ~ pp(aa_set_nat_bool(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5479_mem__case__prodE,axiom,
    ! [Z: int,C2: fun_in7040156458970095540et_int,P: product_prod_int_int] :
      ( pp(aa_set_int_bool(member_int(Z),produc73460835934605544et_int(C2,P)))
     => ~ ! [X4: int,Y3: int] :
            ( ( P = product_Pair_int_int(X4,Y3) )
           => ~ pp(aa_set_int_bool(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5480_mem__case__prodE,axiom,
    ! [Z: set_nat,C2: fun_na2051478540833925974et_nat,P: product_prod_nat_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(Z),produc8404753619367356554et_nat(C2,P)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_set_set_nat_bool(member_set_nat(Z),aa_nat_set_set_nat(aa_nat4995987175846560023et_nat(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5481_mem__case__prodE,axiom,
    ! [Z: set_nat,C2: fun_in7207726656141197838et_nat,P: product_prod_int_int] :
      ( pp(aa_set_set_nat_bool(member_set_nat(Z),produc5233655623923918146et_nat(C2,P)))
     => ~ ! [X4: int,Y3: int] :
            ( ( P = product_Pair_int_int(X4,Y3) )
           => ~ pp(aa_set_set_nat_bool(member_set_nat(Z),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5482_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C2: fun_na1790740735814032627at_nat,P: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc8197505143624133779at_nat(C2,P)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_nat7062706319057040595at_nat(aa_nat7495903657192774236at_nat(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5483_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C2: fun_in1904380326158164027at_nat,P: product_prod_int_int] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),produc1656060378719767003at_nat(C2,P)))
     => ~ ! [X4: int,Y3: int] :
            ( ( P = product_Pair_int_int(X4,Y3) )
           => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Z),aa_int8822742982810151799at_nat(aa_int6362027936895332260at_nat(C2,X4),Y3))) ) ) ).

% mem_case_prodE
tff(fact_5484_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,Y)) ).

% of_int_max
tff(fact_5485_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_rat_rat(ord_max_rat(aa_int_rat(ring_1_of_int_rat,X)),aa_int_rat(ring_1_of_int_rat,Y)) ).

% of_int_max
tff(fact_5486_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,X)),aa_int_int(ring_1_of_int_int,Y)) ).

% of_int_max
tff(fact_5487_case__prodE,axiom,
    ! [C2: fun_Pr248683899324676269t_bool,P: produc859450856879609959at_nat] :
      ( pp(aa_Pro6098352494050694700t_bool(produc6738493412065443883t_bool(C2),P))
     => ~ ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P = produc6161850002892822231at_nat(X4,Y3) )
           => ~ pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(C2,X4),Y3)) ) ) ).

% case_prodE
tff(fact_5488_case__prodE,axiom,
    ! [C2: fun_se5010674451832123565t_bool,P: produc1319942482725812455at_nat] :
      ( pp(aa_Pro4324259192382070444t_bool(produc3930200643136266155t_bool(C2),P))
     => ~ ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
            ( ( P = produc9060074326276436823at_nat(X4,Y3) )
           => ~ pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(C2,X4),Y3)) ) ) ).

% case_prodE
tff(fact_5489_case__prodE,axiom,
    ! [C2: fun_se1534075524646452397t_bool,P: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(produc4641332057840980011t_bool(C2),P))
     => ~ ! [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P = produc2922128104949294807at_nat(X4,Y3) )
           => ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(C2,X4),Y3)) ) ) ).

% case_prodE
tff(fact_5490_case__prodE,axiom,
    ! [C2: fun_nat_fun_nat_bool,P: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(C2),P))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(C2,X4),Y3)) ) ) ).

% case_prodE
tff(fact_5491_case__prodE,axiom,
    ! [C2: fun_int_fun_int_bool,P: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(C2),P))
     => ~ ! [X4: int,Y3: int] :
            ( ( P = product_Pair_int_int(X4,Y3) )
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(C2,X4),Y3)) ) ) ).

% case_prodE
tff(fact_5492_case__prodD,axiom,
    ! [F: fun_Pr248683899324676269t_bool,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( pp(aa_Pro6098352494050694700t_bool(produc6738493412065443883t_bool(F),produc6161850002892822231at_nat(A,B)))
     => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(F,A),B)) ) ).

% case_prodD
tff(fact_5493_case__prodD,axiom,
    ! [F: fun_se5010674451832123565t_bool,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( pp(aa_Pro4324259192382070444t_bool(produc3930200643136266155t_bool(F),produc9060074326276436823at_nat(A,B)))
     => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(F,A),B)) ) ).

% case_prodD
tff(fact_5494_case__prodD,axiom,
    ! [F: fun_se1534075524646452397t_bool,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(produc4641332057840980011t_bool(F),produc2922128104949294807at_nat(A,B)))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(F,A),B)) ) ).

% case_prodD
tff(fact_5495_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B)) ) ).

% case_prodD
tff(fact_5496_case__prodD,axiom,
    ! [F: fun_int_fun_int_bool,A: int,B: int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),product_Pair_int_int(A,B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(F,A),B)) ) ).

% case_prodD
tff(fact_5497_case__prodE_H,axiom,
    ! [C2: fun_na4786964484885125043t_bool,P: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(C2),P),Z))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(C2,X4),Y3),Z)) ) ) ).

% case_prodE'
tff(fact_5498_case__prodD_H,axiom,
    ! [R: fun_na4786964484885125043t_bool,A: nat,B: nat,C2: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C2))
     => pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(R,A),B),C2)) ) ).

% case_prodD'
tff(fact_5499_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_5500_Collect__case__prod__mono,axiom,
    ! [A4: fun_nat_fun_nat_bool,B4: fun_nat_fun_nat_bool] :
      ( ord_le4899916226251588169t_bool(A4,B4)
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(A4))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(B4)))) ) ).

% Collect_case_prod_mono
tff(fact_5501_Collect__case__prod__mono,axiom,
    ! [A4: fun_int_fun_int_bool,B4: fun_int_fun_int_bool] :
      ( ord_le4546832483993983377t_bool(A4,B4)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc8560705615207772305t_bool(A4)),collec213857154873943460nt_int(produc8560705615207772305t_bool(B4))) ) ).

% Collect_case_prod_mono
tff(fact_5502_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_5503_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X))) ).

% tanh_real_gt_neg1
tff(fact_5504_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% sqrt2_less_2
tff(fact_5505_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_nonneg
tff(fact_5506_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_nonneg
tff(fact_5507_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_nonneg
tff(fact_5508_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_pos
tff(fact_5509_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_pos
tff(fact_5510_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_pos
tff(fact_5511_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),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] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_5512_floor__exists1,axiom,
    ! [X: rat] :
    ? [X4: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X4)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),X))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_5513_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_5514_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_5515_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)) ).

% of_int_neg_numeral
tff(fact_5516_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2)) ).

% of_int_neg_numeral
tff(fact_5517_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2)) ).

% of_int_neg_numeral
tff(fact_5518_int__le__real__less,axiom,
    ! [N: int,M2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M2)),one_one_real))) ) ).

% int_le_real_less
tff(fact_5519_int__less__real__le,axiom,
    ! [N: int,M2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M2))) ) ).

% int_less_real_le
tff(fact_5520_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_5521_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_5522_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_5523_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_5524_divmod__nat__def,axiom,
    ! [M2: nat,N: nat] : divmod_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ).

% divmod_nat_def
tff(fact_5525_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_5526_round__unique,axiom,
    ! [X: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_5527_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_5528_of__int__round__gt,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_gt
tff(fact_5529_of__int__round__gt,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_gt
tff(fact_5530_of__int__round__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_ge
tff(fact_5531_of__int__round__ge,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_ge
tff(fact_5532_of__int__round__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_5533_of__int__round__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_5534_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_fn(D))) ).

% int_ge_less_than_def
tff(fact_5535_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_fo(D))) ).

% int_ge_less_than2_def
tff(fact_5536_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_cf,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M2),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M2),aa_int_int(aa_int_fun_int_int(minus_minus_int,M2),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_5537_split__part,axiom,
    ! [P2: bool,Q: fun_nat_fun_nat_bool,X3: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_fp(P2),Q)),X3))
    <=> ( pp(P2)
        & pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(Q),X3)) ) ) ).

% split_part
tff(fact_5538_split__part,axiom,
    ! [P2: bool,Q: fun_int_fun_int_bool,X3: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_fq(P2),Q)),X3))
    <=> ( pp(P2)
        & pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(Q),X3)) ) ) ).

% split_part
tff(fact_5539_sum_Oneutral__const,axiom,
    ! [A4: set_int] : groups4538972089207619220nt_int(aTP_Lamm_ca,A4) = zero_zero_int ).

% sum.neutral_const
tff(fact_5540_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bz),A4) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5541_sum_Oneutral__const,axiom,
    ! [A4: set_complex] : groups7754918857620584856omplex(aTP_Lamm_bw,A4) = zero_zero_complex ).

% sum.neutral_const
tff(fact_5542_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fr),A4) = zero_zero_real ).

% sum.neutral_const
tff(fact_5543_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_5544_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_5545_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_5546_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_5547_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_5548_sum_Oempty,axiom,
    ! [G: fun_nat_complex] : groups2073611262835488442omplex(G,bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_5549_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_5550_sum_Oempty,axiom,
    ! [G: fun_nat_int] : aa_set_nat_int(groups3539618377306564664at_int(G),bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_5551_sum_Oempty,axiom,
    ! [G: fun_int_complex] : groups3049146728041665814omplex(G,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_5552_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_5553_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),F3))
           => ( aa_int_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5554_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),F3))
           => ( aa_complex_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5555_sum__eq__0__iff,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( ( groups977919841031483927at_nat(F,F3) = zero_zero_nat )
      <=> ! [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),F3))
           => ( aa_Pro4243173770044397973at_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5556_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),F3))
           => ( aa_nat_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5557_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2073611262835488442omplex(G,A4) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5558_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3049146728041665814omplex(G,A4) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5559_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5560_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5561_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2906978787729119204at_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5562_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5563_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5564_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups4541462559716669496nt_nat(G,A4) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5565_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5693394587270226106ex_nat(G,A4) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5566_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_int] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),A4) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_5567_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5568_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5569_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_fs(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_fs(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5570_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ft(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ft(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5571_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_fu(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_fu(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5572_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fv(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fv(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5573_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fw(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fw(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5574_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fx(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fx(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5575_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fy(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fy(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5576_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fz(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fz(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5577_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ga(A),B),S2) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ga(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5578_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gb(A),B),S2) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gb(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5579_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gc(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gc(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5580_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gd(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gd(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5581_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ge(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ge(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5582_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gf(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gf(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5583_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gg(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gg(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5584_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gh(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gh(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5585_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gi(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gi(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5586_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gj(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gj(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5587_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gk(A),B),S2) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gk(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5588_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gl(A),B),S2) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gl(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5589_of__int__sum,axiom,
    ! [F: fun_complex_int,A4: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A4)) = groups7754918857620584856omplex(aTP_Lamm_gm(F),A4) ).

% of_int_sum
tff(fact_5590_of__int__sum,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_real(ring_1_of_int_real,aa_set_nat_int(groups3539618377306564664at_int(F),A4)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(F)),A4) ).

% of_int_sum
tff(fact_5591_of__int__sum,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A4)) = groups8778361861064173332t_real(aTP_Lamm_go(F),A4) ).

% of_int_sum
tff(fact_5592_of__int__sum,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A4)) = groups3906332499630173760nt_rat(aTP_Lamm_gp(F),A4) ).

% of_int_sum
tff(fact_5593_of__int__sum,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A4)) = groups4538972089207619220nt_int(aTP_Lamm_gq(F),A4) ).

% of_int_sum
tff(fact_5594_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] : pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(aTP_Lamm_gr),Prod)) ).

% prod.disc_eq_case
tff(fact_5595_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_gs),Prod)) ).

% prod.disc_eq_case
tff(fact_5596_sum_Oneutral,axiom,
    ! [A4: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => ( aa_int_int(G,X4) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A4) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_5597_sum_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_5598_sum_Oneutral,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A4))
         => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A4) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_5599_sum_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => ( aa_nat_real(G,X4) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A4) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_5600_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A4: set_real] :
      ( ( groups5754745047067104278omplex(G,A4) != zero_zero_complex )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ( aa_real_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5601_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A4: set_nat] :
      ( ( groups2073611262835488442omplex(G,A4) != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A4))
           => ( aa_nat_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5602_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A4: set_int] :
      ( ( groups3049146728041665814omplex(G,A4) != zero_zero_complex )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ( aa_int_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5603_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A4: set_real] :
      ( ( groups8097168146408367636l_real(G,A4) != zero_zero_real )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5604_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A4: set_int] :
      ( ( groups8778361861064173332t_real(G,A4) != zero_zero_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5605_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A4: set_real] :
      ( ( groups1300246762558778688al_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5606_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A4: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A4))
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5607_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A4: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ( aa_int_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5608_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A4: set_real] :
      ( ( groups1935376822645274424al_nat(G,A4) != zero_zero_nat )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ( aa_real_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5609_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A4: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A4) != zero_zero_nat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ( aa_int_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5610_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5))) ) ).

% sum_mono
tff(fact_5611_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5))) ) ).

% sum_mono
tff(fact_5612_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5))) ) ).

% sum_mono
tff(fact_5613_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5))) ) ).

% sum_mono
tff(fact_5614_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5))) ) ).

% sum_mono
tff(fact_5615_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5))) ) ).

% sum_mono
tff(fact_5616_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),K5)),aa_set_nat_int(groups3539618377306564664at_int(G),K5))) ) ).

% sum_mono
tff(fact_5617_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,I3)),aa_int_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5))) ) ).

% sum_mono
tff(fact_5618_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I3)),aa_nat_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),K5)),aa_set_nat_nat(groups3542108847815614940at_nat(G),K5))) ) ).

% sum_mono
tff(fact_5619_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),aa_nat_real(G,I3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),K5)),aa_set_nat_real(groups6591440286371151544t_real(G),K5))) ) ).

% sum_mono
tff(fact_5620_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gt(G),H),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A4)),groups4538972089207619220nt_int(H,A4)) ).

% sum.distrib
tff(fact_5621_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gu(G),H)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A4)) ).

% sum.distrib
tff(fact_5622_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A4: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_gv(G),H),A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A4)),groups7754918857620584856omplex(H,A4)) ).

% sum.distrib
tff(fact_5623_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A4: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gw(G),H)),A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),A4)),aa_set_nat_real(groups6591440286371151544t_real(H),A4)) ).

% sum.distrib
tff(fact_5624_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_int,G: fun_real_fun_int_int,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( groups1932886352136224148al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_gy(B4),G),R),A4) = groups4538972089207619220nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_ha(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5625_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun3504929462025808522at_int(aa_fun3325927984641084153at_int(aTP_Lamm_hc(B4),G),R)),A4) = groups4538972089207619220nt_int(aa_fun4504283173926863078nt_int(aa_fun5512849559334105941nt_int(aTP_Lamm_he(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5626_sum_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_int,G: fun_co333780937270390912nt_int,R: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( groups5690904116761175830ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_hg(B4),G),R),A4) = groups4538972089207619220nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_hj(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5627_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_hl(B4),G),R),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_hn(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5628_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_hp(B4),G),R),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_hr(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5629_sum_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_ht(B4),G),R),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_hv(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5630_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_complex,G: fun_re7208826350674946562omplex,R: fun_re3010880487650781423x_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_hx(B4),G),R),A4) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_hz(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5631_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_complex,G: fun_na1591161823130732966omplex,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => ( groups2073611262835488442omplex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_ib(B4),G),R),A4) = groups7754918857620584856omplex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_id(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5632_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5750378413879446255x_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => ( groups3049146728041665814omplex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_if(B4),G),R),A4) = groups7754918857620584856omplex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_ih(A4),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5633_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_re4238438383274954274t_real,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups8097168146408367636l_real(aa_fun7981600115099051494l_real(aa_fun2847282595593119957l_real(aTP_Lamm_ii(B4),G),R),A4) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun555827300007793674t_real(aa_fun8353879187736340217t_real(aTP_Lamm_ik(A4),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_5634_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5635_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5636_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5637_sum__nonpos,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5638_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A4)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5639_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5640_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A4)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5641_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,A4)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_5642_sum__nonpos,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_5643_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4538972089207619220nt_int(F,A4)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_5644_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A4))) ) ).

% sum_nonneg
tff(fact_5645_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A4))) ) ).

% sum_nonneg
tff(fact_5646_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A4))) ) ).

% sum_nonneg
tff(fact_5647_sum__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A4))) ) ).

% sum_nonneg
tff(fact_5648_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A4))) ) ).

% sum_nonneg
tff(fact_5649_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A4))) ) ).

% sum_nonneg
tff(fact_5650_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A4))) ) ).

% sum_nonneg
tff(fact_5651_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1932886352136224148al_int(F,A4))) ) ).

% sum_nonneg
tff(fact_5652_sum__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups3539618377306564664at_int(F),A4))) ) ).

% sum_nonneg
tff(fact_5653_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(F,A4))) ) ).

% sum_nonneg
tff(fact_5654_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I6: set_real,G: fun_real_rat,I2: real] :
      ( ( groups1300246762558778688al_rat(F,I6) = groups1300246762558778688al_rat(G,I6) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) )
       => ( pp(aa_set_real_bool(member_real(I2),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_rat(F,I2) = aa_real_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5655_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I6: set_nat,G: fun_nat_rat,I2: nat] :
      ( ( groups2906978787729119204at_rat(F,I6) = groups2906978787729119204at_rat(G,I6) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) )
       => ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
           => ( aa_nat_rat(F,I2) = aa_nat_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5656_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I6: set_int,G: fun_int_rat,I2: int] :
      ( ( groups3906332499630173760nt_rat(F,I6) = groups3906332499630173760nt_rat(G,I6) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) )
       => ( pp(aa_set_int_bool(member_int(I2),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_rat(F,I2) = aa_int_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5657_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I6: set_complex,G: fun_complex_rat,I2: complex] :
      ( ( groups5058264527183730370ex_rat(F,I6) = groups5058264527183730370ex_rat(G,I6) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I2),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_rat(F,I2) = aa_complex_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5658_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I2: real] :
      ( ( groups1935376822645274424al_nat(F,I6) = groups1935376822645274424al_nat(G,I6) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) )
       => ( pp(aa_set_real_bool(member_real(I2),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_nat(F,I2) = aa_real_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5659_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I2: int] :
      ( ( groups4541462559716669496nt_nat(F,I6) = groups4541462559716669496nt_nat(G,I6) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) )
       => ( pp(aa_set_int_bool(member_int(I2),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_nat(F,I2) = aa_int_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5660_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I2: complex] :
      ( ( groups5693394587270226106ex_nat(F,I6) = groups5693394587270226106ex_nat(G,I6) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I2),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_nat(F,I2) = aa_complex_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5661_sum__mono__inv,axiom,
    ! [F: fun_real_int,I6: set_real,G: fun_real_int,I2: real] :
      ( ( groups1932886352136224148al_int(F,I6) = groups1932886352136224148al_int(G,I6) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3))) )
       => ( pp(aa_set_real_bool(member_real(I2),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_int(F,I2) = aa_real_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5662_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I6: set_nat,G: fun_nat_int,I2: nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(F),I6) = aa_set_nat_int(groups3539618377306564664at_int(G),I6) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3))) )
       => ( pp(aa_set_nat_bool(member_nat(I2),I6))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
           => ( aa_nat_int(F,I2) = aa_nat_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5663_sum__mono__inv,axiom,
    ! [F: fun_complex_int,I6: set_complex,G: fun_complex_int,I2: complex] :
      ( ( groups5690904116761175830ex_int(F,I6) = groups5690904116761175830ex_int(G,I6) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,I3)),aa_complex_int(G,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I2),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_int(F,I2) = aa_complex_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5664_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_complex,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_il(A4),P2))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_im(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5665_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_complex,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_in(A4),P2))) = groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_io(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5666_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_complex,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ip(A4),P2))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_iq(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5667_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_real,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_il(A4),P2))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_ir(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5668_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_real,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ip(A4),P2))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_is(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5669_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_real,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_it(A4),P2))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_iu(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5670_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_rat,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_il(A4),P2))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_iv(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5671_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_rat,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_in(A4),P2))) = groups2906978787729119204at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_iw(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5672_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_rat,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ip(A4),P2))) = groups3906332499630173760nt_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_ix(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5673_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_rat,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_it(A4),P2))) = groups5058264527183730370ex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_iy(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5674_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( ( groups8097168146408367636l_real(F,A4) = zero_zero_real )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A4))
             => ( aa_real_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5675_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( ( groups8778361861064173332t_real(F,A4) = zero_zero_real )
        <=> ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A4))
             => ( aa_int_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5676_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( ( groups5808333547571424918x_real(F,A4) = zero_zero_real )
        <=> ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A4))
             => ( aa_complex_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5677_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( ( groups1300246762558778688al_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A4))
             => ( aa_real_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5678_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( ( groups2906978787729119204at_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A4))
             => ( aa_nat_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5679_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( ( groups3906332499630173760nt_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A4))
             => ( aa_int_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5680_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
       => ( ( groups5058264527183730370ex_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A4))
             => ( aa_complex_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5681_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
       => ( ( groups1935376822645274424al_nat(F,A4) = zero_zero_nat )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A4))
             => ( aa_real_nat(F,X5) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5682_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
       => ( ( groups4541462559716669496nt_nat(F,A4) = zero_zero_nat )
        <=> ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A4))
             => ( aa_int_nat(F,X5) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5683_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
       => ( ( groups5693394587270226106ex_nat(F,A4) = zero_zero_nat )
        <=> ! [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A4))
             => ( aa_complex_nat(F,X5) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5684_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_real,I2: fun_int_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I2,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5685_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_real,I2: fun_complex_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I2,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_complex_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5686_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_int_real,I2: fun_int_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I2,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_int_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5687_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_complex_real,I2: fun_complex_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_complex(I2,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5688_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_rat,I2: fun_nat_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I2,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5689_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_rat,I2: fun_int_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_nat(I2,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_int_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5690_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: fun_complex_rat,I2: fun_complex_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_nat(I2,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_complex_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5691_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_nat_rat,I2: fun_nat_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_int(I2,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5692_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_rat,I2: fun_int_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I2,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5693_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_rat,I2: fun_complex_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I2,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_complex_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5694_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5695_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
       => ( ? [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5696_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
       => ( ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A4))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5697_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A4))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5698_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
       => ( ? [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A4))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5699_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5700_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
       => ( ? [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5701_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,X4))) )
       => ( ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A4))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(G,X3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),aa_set_nat_int(groups3539618377306564664at_int(G),A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5702_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,X4))) )
       => ( ? [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A4))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,X3)),aa_complex_int(G,X3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5703_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(G,X4))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A4))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(F,X3)),aa_int_int(G,X3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(G,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5704_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X22))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups2073611262835488442omplex(H,S2)),groups2073611262835488442omplex(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5705_sum_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X22))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_int_complex(H,X4)),aa_int_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups3049146728041665814omplex(H,S2)),groups3049146728041665814omplex(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5706_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5707_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5708_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5709_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_int_rat(H,X4)),aa_int_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5710_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5711_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X15),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5712_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X15),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_5713_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_bool,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R,zero_zero_int),zero_zero_int))
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X15),X22))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y15),Y23)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_nat_int(H,X4)),aa_nat_int(G,X4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_set_nat_int(groups3539618377306564664at_int(H),S2)),aa_set_nat_int(groups3539618377306564664at_int(G),S2))) ) ) ) ) ).

% sum.related
tff(fact_5714_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5715_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5716_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5717_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5718_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5719_sum__strict__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5720_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5721_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5722_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5723_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4))) ) ) ) ).

% sum_strict_mono
tff(fact_5724_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I2: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I2,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_real_real(J,aa_real_real(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5725_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S2: set_real,I2: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I2,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T2,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                 => ( aa_real_int(J,aa_int_real(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5726_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_real,S2: set_int,I2: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I2,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_int_real(J,aa_real_int(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5727_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_int,S2: set_int,I2: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_int_int(I2,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T2,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                 => ( aa_int_int(J,aa_int_int(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5728_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I2: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I2,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_real_real(J,aa_real_real(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5729_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S2: set_real,I2: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_real,H: fun_int_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I2,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T2,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                 => ( aa_real_int(J,aa_int_real(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_real(H,aa_real_int(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5730_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_complex,S2: set_real,I2: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_complex_real(I2,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T2,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                 => ( aa_real_complex(J,aa_complex_real(I2,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_complex_real(H,aa_real_complex(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5731_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_real,S2: set_int,I2: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_real,H: fun_real_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I2,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_int_real(J,aa_real_int(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_real(H,aa_int_real(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5732_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_int,S2: set_int,I2: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_int_int(I2,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T2,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                 => ( aa_int_int(J,aa_int_int(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_int_real(H,aa_int_int(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5733_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_complex,S2: set_int,I2: fun_complex_int,J: fun_int_complex,T2: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_complex_int(I2,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T2,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                 => ( aa_int_complex(J,aa_complex_int(I2,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_complex_real(H,aa_int_complex(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5734_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y))) ) ).

% round_mono
tff(fact_5735_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B4: real,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
       => ( ( groups8097168146408367636l_real(F,S) = B4 )
         => ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5736_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B4: real,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
       => ( ( groups8778361861064173332t_real(F,S) = B4 )
         => ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5737_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B4: real,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( ( groups5808333547571424918x_real(F,S) = B4 )
         => ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5738_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B4: rat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
       => ( ( groups1300246762558778688al_rat(F,S) = B4 )
         => ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5739_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B4: rat,I2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
       => ( ( groups2906978787729119204at_rat(F,S) = B4 )
         => ( pp(aa_set_nat_bool(member_nat(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5740_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B4: rat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = B4 )
         => ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5741_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B4: rat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = B4 )
         => ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5742_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B4: nat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
       => ( ( groups1935376822645274424al_nat(F,S) = B4 )
         => ( pp(aa_set_real_bool(member_real(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5743_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_nat,B4: nat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3))) )
       => ( ( groups4541462559716669496nt_nat(F,S) = B4 )
         => ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5744_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_nat,B4: nat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3))) )
       => ( ( groups5693394587270226106ex_nat(F,S) = B4 )
         => ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5745_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( pp(aa_set_real_bool(member_real(I2),S))
           => ( aa_real_real(F,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5746_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I2),S))
           => ( aa_int_real(F,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5747_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I2),S))
           => ( aa_complex_real(F,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5748_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
       => ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_real_bool(member_real(I2),S))
           => ( aa_real_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5749_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
       => ( ( groups2906978787729119204at_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_nat_bool(member_nat(I2),S))
           => ( aa_nat_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5750_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_int_bool(member_int(I2),S))
           => ( aa_int_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5751_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_complex_bool(member_complex(I2),S))
           => ( aa_complex_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5752_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_real_bool(member_real(I2),S))
           => ( aa_real_nat(F,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5753_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_nat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3))) )
       => ( ( groups4541462559716669496nt_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_int_bool(member_int(I2),S))
           => ( aa_int_nat(F,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5754_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_nat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3))) )
       => ( ( groups5693394587270226106ex_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_complex_bool(member_complex(I2),S))
           => ( aa_complex_nat(F,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5755_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_iz(G)))) = groups5754745047067104278omplex(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5756_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_ja(G)))) = groups3049146728041665814omplex(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5757_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_jb(G)))) = groups8097168146408367636l_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5758_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_jc(G)))) = groups8778361861064173332t_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5759_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_jd(G)))) = groups5808333547571424918x_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5760_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_je(G)))) = groups1300246762558778688al_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5761_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_jf(G)))) = groups3906332499630173760nt_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5762_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_jg(G)))) = groups5058264527183730370ex_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5763_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_jh(G)))) = groups1935376822645274424al_nat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5764_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_ji(G)))) = groups4541462559716669496nt_nat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5765_sum__pos2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I2),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5766_sum__pos2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I2),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5767_sum__pos2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I2),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5768_sum__pos2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5769_sum__pos2,axiom,
    ! [I6: set_nat,I2: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5770_sum__pos2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5771_sum__pos2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5772_sum__pos2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I2),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I6))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5773_sum__pos2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I2),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I6))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5774_sum__pos2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I2),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I6))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_5775_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5776_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5777_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5778_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5779_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5780_sum__pos,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5781_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5782_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5783_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5784_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_5785_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,T2) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5786_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5787_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5788_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5789_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5790_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5791_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5792_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups1932886352136224148al_int(G,T2) = groups1932886352136224148al_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5793_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5794_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => ( aa_nat_complex(G,X4) = aa_nat_complex(H,X4) ) )
           => ( groups2073611262835488442omplex(G,T2) = groups2073611262835488442omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5795_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5796_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5797_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5798_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5799_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5800_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5801_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5802_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_int(H,X4) = zero_zero_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups1932886352136224148al_int(G,S2) = groups1932886352136224148al_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5803_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(H,X4) = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5804_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_nat,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => ( aa_nat_complex(G,X4) = aa_nat_complex(H,X4) ) )
           => ( groups2073611262835488442omplex(G,S2) = groups2073611262835488442omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5805_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5806_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5807_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5808_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5809_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T2) = groups2073611262835488442omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5810_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5811_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),T2) = aa_set_nat_int(groups3539618377306564664at_int(G),S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5812_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5813_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5814_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5815_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5816_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5817_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5818_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5819_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S2) = groups2073611262835488442omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5820_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5821_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),S2) = aa_set_nat_int(groups3539618377306564664at_int(G),T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5822_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5823_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5824_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5825_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G,A4) = groups5754745047067104278omplex(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5826_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5827_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5828_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5829_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5830_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5831_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) )
               => ( groups5693394587270226106ex_nat(G,A4) = groups5693394587270226106ex_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5832_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) )
               => ( groups1932886352136224148al_int(G,A4) = groups1932886352136224148al_int(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5833_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) )
               => ( groups5690904116761175830ex_int(G,A4) = groups5690904116761175830ex_int(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5834_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B4: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,C4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),C4))
       => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),C4))
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),minus_minus_set_nat(C4,A4)))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(C4,B4)))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups2073611262835488442omplex(G,C4) = groups2073611262835488442omplex(H,C4) )
               => ( groups2073611262835488442omplex(G,A4) = groups2073611262835488442omplex(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5835_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A4) = groups5754745047067104278omplex(H,B4) )
              <=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5836_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5837_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5838_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5839_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5840_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5841_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A4) = groups5693394587270226106ex_nat(H,B4) )
              <=> ( groups5693394587270226106ex_nat(G,C4) = groups5693394587270226106ex_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5842_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_int(H,B3) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A4) = groups1932886352136224148al_int(H,B4) )
              <=> ( groups1932886352136224148al_int(G,C4) = groups1932886352136224148al_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5843_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_int(H,B3) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A4) = groups5690904116761175830ex_int(H,B4) )
              <=> ( groups5690904116761175830ex_int(G,C4) = groups5690904116761175830ex_int(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5844_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B4: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,C4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),C4))
       => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),C4))
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),minus_minus_set_nat(C4,A4)))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(C4,B4)))
                 => ( aa_nat_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups2073611262835488442omplex(G,A4) = groups2073611262835488442omplex(H,B4) )
              <=> ( groups2073611262835488442omplex(G,C4) = groups2073611262835488442omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5845_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5846_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,B4))),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5847_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,B4))),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5848_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5849_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),minus_minus_set_nat(A4,B4))),aa_set_nat_int(groups3539618377306564664at_int(G),B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5850_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( groups2906978787729119204at_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A4,B4))),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5851_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups4541462559716669496nt_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A4,B4))),groups4541462559716669496nt_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5852_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups8778361861064173332t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A4,B4))),groups8778361861064173332t_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5853_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups3906332499630173760nt_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A4,B4))),groups3906332499630173760nt_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5854_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups4538972089207619220nt_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A4,B4))),groups4538972089207619220nt_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_5855_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5856_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5857_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,B4)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5858_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
       => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),aa_set_nat_int(groups3539618377306564664at_int(F),B4)) ) ) ) ).

% sum_diff
tff(fact_5859_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A4,B4)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5860_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5861_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A4,B4)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5862_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5863_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A4,B4)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A4)),groups7754918857620584856omplex(F,B4)) ) ) ) ).

% sum_diff
tff(fact_5864_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A4)),aa_set_nat_real(groups6591440286371151544t_real(F),B4)) ) ) ) ).

% sum_diff
tff(fact_5865_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5866_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5867_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5868_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5869_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B4,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5870_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5871_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A4)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5872_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,A4)),groups1932886352136224148al_int(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5873_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A4)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_5874_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B4,A4)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),aa_set_nat_int(groups3539618377306564664at_int(F),B4))) ) ) ) ).

% sum_mono2
tff(fact_5875_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B4,A4)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5876_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B4,A4)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5877_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B4,A4)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5878_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B4,A4)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5879_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( pp(aa_set_nat_bool(member_nat(B),minus_minus_set_nat(B4,A4)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B)))
           => ( ! [X4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X4),B4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5880_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B4,A4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B4))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5881_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B4,A4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B4))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5882_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B4,A4)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B4))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F,A4)),groups1932886352136224148al_int(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5883_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B4,A4)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_complex_int(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B4))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5884_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( pp(aa_set_nat_bool(member_nat(B),minus_minus_set_nat(B4,A4)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,B)))
           => ( ! [X4: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X4),B4))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),aa_set_nat_int(groups3539618377306564664at_int(F),B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5885_mask__numeral,axiom,
    ! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_5886_mask__numeral,axiom,
    ! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,N)))) ).

% mask_numeral
tff(fact_5887_set__encode__insert,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A4))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_5888_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = 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,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_5889_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_5890_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_5891_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% ceiling_less_neg_numeral
tff(fact_5892_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% ceiling_less_neg_numeral
tff(fact_5893_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jj(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = 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,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_5894_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_5895_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) ).

% singletonI
tff(fact_5896_singletonI,axiom,
    ! [A: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat))) ).

% singletonI
tff(fact_5897_singletonI,axiom,
    ! [A: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) ).

% singletonI
tff(fact_5898_singletonI,axiom,
    ! [A: real] : pp(aa_set_real_bool(member_real(A),insert_real(A,bot_bot_set_real))) ).

% singletonI
tff(fact_5899_singletonI,axiom,
    ! [A: nat] : pp(aa_set_nat_bool(member_nat(A),insert_nat(A,bot_bot_set_nat))) ).

% singletonI
tff(fact_5900_singletonI,axiom,
    ! [A: int] : pp(aa_set_int_bool(member_int(A),insert_int(A,bot_bot_set_int))) ).

% singletonI
tff(fact_5901_finite__insert,axiom,
    ! [A: real,A4: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,insert_real(A,A4)))
    <=> pp(aa_set_real_bool(finite_finite_real,A4)) ) ).

% finite_insert
tff(fact_5902_finite__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A4)))
    <=> pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4)) ) ).

% finite_insert
tff(fact_5903_finite__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A4)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ).

% finite_insert
tff(fact_5904_finite__insert,axiom,
    ! [A: int,A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,insert_int(A,A4)))
    <=> pp(aa_set_int_bool(finite_finite_int,A4)) ) ).

% finite_insert
tff(fact_5905_finite__insert,axiom,
    ! [A: complex,A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A4)))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ).

% finite_insert
tff(fact_5906_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A4)))
    <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4)) ) ).

% finite_insert
tff(fact_5907_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(aTP_Lamm_jk(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv
tff(fact_5908_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_jl(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_5909_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jm,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_5910_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_jn,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv
tff(fact_5911_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jo,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_5912_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jp,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_5913_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(aTP_Lamm_jq,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_5914_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(fequal7949547063978738364at_nat(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv2
tff(fact_5915_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_5916_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(fequal_set_nat,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_5917_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv2
tff(fact_5918_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_5919_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_5920_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_5921_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,A4) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
    <=> ( ( A = B )
        & pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5922_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,A4) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
    <=> ( ( A = B )
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5923_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A4: set_real,B: real] :
      ( ( insert_real(A,A4) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A4),insert_real(B,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5924_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( insert_nat(A,A4) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),insert_nat(B,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5925_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A4: set_int,B: int] :
      ( ( insert_int(A,A4) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A4),insert_int(B,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5926_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(A,A4) )
    <=> ( ( A = B )
        & pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5927_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A,A4) )
    <=> ( ( A = B )
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5928_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A4: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A4) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A4),insert_real(B,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5929_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A4) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),insert_nat(B,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5930_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A4: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A4) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A4),insert_int(B,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5931_atLeastAtMost__singleton,axiom,
    ! [A: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_5932_atLeastAtMost__singleton,axiom,
    ! [A: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_5933_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_5934_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) = insert_nat(C2,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5935_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = insert_int(C2,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5936_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real(C2,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5937_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = insert9069300056098147895at_nat(A,A4) ).

% insert_Diff_single
tff(fact_5938_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A,A4) ).

% insert_Diff_single
tff(fact_5939_insert__Diff__single,axiom,
    ! [A: real,A4: set_real] : insert_real(A,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = insert_real(A,A4) ).

% insert_Diff_single
tff(fact_5940_insert__Diff__single,axiom,
    ! [A: int,A4: set_int] : insert_int(A,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = insert_int(A,A4) ).

% insert_Diff_single
tff(fact_5941_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A4) ).

% insert_Diff_single
tff(fact_5942_finite__Diff__insert,axiom,
    ! [A4: set_real,A: real,B4: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A4,insert_real(A,B4))))
    <=> pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A4,B4))) ) ).

% finite_Diff_insert
tff(fact_5943_finite__Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,B4))))
    <=> pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A4,B4))) ) ).

% finite_Diff_insert
tff(fact_5944_finite__Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A4,insert_int(A,B4))))
    <=> pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A4,B4))) ) ).

% finite_Diff_insert
tff(fact_5945_finite__Diff__insert,axiom,
    ! [A4: set_complex,A: complex,B4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,insert_complex(A,B4))))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4))) ) ).

% finite_Diff_insert
tff(fact_5946_finite__Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,B4))))
    <=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A4,B4))) ) ).

% finite_Diff_insert
tff(fact_5947_finite__Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A4,insert_nat(A,B4))))
    <=> pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A4,B4))) ) ).

% finite_Diff_insert
tff(fact_5948_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_5949_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_5950_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_5951_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_5952_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,N) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_5953_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,N) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_5954_sum_Oinsert,axiom,
    ! [A4: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1935376822645274424al_nat(G,insert_real(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5955_sum_Oinsert,axiom,
    ! [A4: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( groups4541462559716669496nt_nat(G,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5956_sum_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups5693394587270226106ex_nat(G,insert_complex(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5957_sum_Oinsert,axiom,
    ! [A4: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1932886352136224148al_int(G,insert_real(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5958_sum_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,X)),aa_set_nat_int(groups3539618377306564664at_int(G),A4)) ) ) ) ).

% sum.insert
tff(fact_5959_sum_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups5690904116761175830ex_int(G,insert_complex(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5960_sum_Oinsert,axiom,
    ! [A4: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups8097168146408367636l_real(G,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5961_sum_Oinsert,axiom,
    ! [A4: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( groups8778361861064173332t_real(G,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5962_sum_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups5808333547571424918x_real(G,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5963_sum_Oinsert,axiom,
    ! [A4: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1300246762558778688al_rat(G,insert_real(X,A4)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_5964_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_5965_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_5966_subset__Compl__singleton,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),uminus935396558254630718at_nat(insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))))
    <=> ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),A4)) ) ).

% subset_Compl_singleton
tff(fact_5967_subset__Compl__singleton,axiom,
    ! [A4: set_set_nat,B: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat))))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(B),A4)) ) ).

% subset_Compl_singleton
tff(fact_5968_subset__Compl__singleton,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))))
    <=> ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),A4)) ) ).

% subset_Compl_singleton
tff(fact_5969_subset__Compl__singleton,axiom,
    ! [A4: set_real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real))))
    <=> ~ pp(aa_set_real_bool(member_real(B),A4)) ) ).

% subset_Compl_singleton
tff(fact_5970_subset__Compl__singleton,axiom,
    ! [A4: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat))))
    <=> ~ pp(aa_set_nat_bool(member_nat(B),A4)) ) ).

% subset_Compl_singleton
tff(fact_5971_subset__Compl__singleton,axiom,
    ! [A4: set_int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int))))
    <=> ~ pp(aa_set_int_bool(member_int(B),A4)) ) ).

% subset_Compl_singleton
tff(fact_5972_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z) ).

% ceiling_add_of_int
tff(fact_5973_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),Z) ).

% ceiling_add_of_int
tff(fact_5974_ceiling__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_le_zero
tff(fact_5975_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_le_zero
tff(fact_5976_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% zero_less_ceiling
tff(fact_5977_zero__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_ceiling
tff(fact_5978_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_numeral
tff(fact_5979_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_numeral
tff(fact_5980_ceiling__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_less_one
tff(fact_5981_ceiling__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_less_one
tff(fact_5982_one__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% one_le_ceiling
tff(fact_5983_one__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_le_ceiling
tff(fact_5984_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_5985_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_5986_ceiling__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% ceiling_le_one
tff(fact_5987_ceiling__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ).

% ceiling_le_one
tff(fact_5988_one__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ).

% one_less_ceiling
tff(fact_5989_one__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ).

% one_less_ceiling
tff(fact_5990_set__replicate,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] :
      ( ( N != zero_zero_nat )
     => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ).

% set_replicate
tff(fact_5991_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_5992_set__replicate,axiom,
    ! [N: nat,X: product_prod_nat_nat] :
      ( ( N != zero_zero_nat )
     => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ).

% set_replicate
tff(fact_5993_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_5994_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_5995_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_5996_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5997_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5998_ceiling__add__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_add_one
tff(fact_5999_ceiling__add__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_add_one
tff(fact_6000_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6001_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6002_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = 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,M2),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6003_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = 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,M2),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6004_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6005_ceiling__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real))) ) ).

% ceiling_less_zero
tff(fact_6006_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ) ).

% ceiling_less_zero
tff(fact_6007_zero__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)) ) ).

% zero_le_ceiling
tff(fact_6008_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)) ) ).

% zero_le_ceiling
tff(fact_6009_sum__zero__power,axiom,
    ! [A4: set_nat,C2: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_jr(C2),A4) = aa_nat_complex(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_jr(C2),A4) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_6010_sum__zero__power,axiom,
    ! [A4: set_nat,C2: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_js(C2),A4) = aa_nat_rat(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_js(C2),A4) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_6011_sum__zero__power,axiom,
    ! [A4: set_nat,C2: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jt(C2)),A4) = aa_nat_real(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jt(C2)),A4) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_6012_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_6013_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% ceiling_less_numeral
tff(fact_6014_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_le_ceiling
tff(fact_6015_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X)) ) ).

% numeral_le_ceiling
tff(fact_6016_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_6017_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_6018_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_6019_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_6020_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ju(C2),D),A4) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C2,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ju(C2),D),A4) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_6021_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: fun_nat_rat,D: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_jv(C2),D),A4) = aa_rat_rat(divide_divide_rat(aa_nat_rat(C2,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_jv(C2),D),A4) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_6022_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jw(C2),D)),A4) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C2,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jw(C2),D)),A4) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_6023_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_6024_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_6025_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_6026_singletonD,axiom,
    ! [B: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B),insert_real(A,bot_bot_set_real)))
     => ( B = A ) ) ).

% singletonD
tff(fact_6027_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B),insert_nat(A,bot_bot_set_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_6028_singletonD,axiom,
    ! [B: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B),insert_int(A,bot_bot_set_int)))
     => ( B = A ) ) ).

% singletonD
tff(fact_6029_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6030_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6031_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6032_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B),insert_real(A,bot_bot_set_real)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6033_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B),insert_nat(A,bot_bot_set_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6034_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B),insert_int(A,bot_bot_set_int)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6035_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C2: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(C2,insert9069300056098147895at_nat(D,bot_bo228742789529271731at_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6036_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C2: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(C2,insert8211810215607154385at_nat(D,bot_bo2099793752762293965at_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6037_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( insert_real(A,insert_real(B,bot_bot_set_real)) = insert_real(C2,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6038_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B,bot_bot_set_nat)) = insert_nat(C2,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6039_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C2,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6040_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,A4) != bot_bo228742789529271731at_nat ).

% insert_not_empty
tff(fact_6041_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,A4) != bot_bo2099793752762293965at_nat ).

% insert_not_empty
tff(fact_6042_insert__not__empty,axiom,
    ! [A: real,A4: set_real] : insert_real(A,A4) != bot_bot_set_real ).

% insert_not_empty
tff(fact_6043_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,A4) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_6044_insert__not__empty,axiom,
    ! [A: int,A4: set_int] : insert_int(A,A4) != bot_bot_set_int ).

% insert_not_empty
tff(fact_6045_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6046_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6047_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( insert_real(A,bot_bot_set_real) = insert_real(B,bot_bot_set_real) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6048_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6049_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_6050_finite_OinsertI,axiom,
    ! [A4: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => pp(aa_set_real_bool(finite_finite_real,insert_real(A,A4))) ) ).

% finite.insertI
tff(fact_6051_finite_OinsertI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4))
     => pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,insert9069300056098147895at_nat(A,A4))) ) ).

% finite.insertI
tff(fact_6052_finite_OinsertI,axiom,
    ! [A4: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A4))) ) ).

% finite.insertI
tff(fact_6053_finite_OinsertI,axiom,
    ! [A4: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => pp(aa_set_int_bool(finite_finite_int,insert_int(A,A4))) ) ).

% finite.insertI
tff(fact_6054_finite_OinsertI,axiom,
    ! [A4: set_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A4))) ) ).

% finite.insertI
tff(fact_6055_finite_OinsertI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,insert8211810215607154385at_nat(A,A4))) ) ).

% finite.insertI
tff(fact_6056_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_6057_sum__diff1__nat,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr1253225528153422666at_nat] :
      ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A4))
       => ( groups3860910324918113789at_nat(F,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups3860910324918113789at_nat(F,A4)),aa_Pro6313114768868464763at_nat(F,A)) ) )
      & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A4))
       => ( groups3860910324918113789at_nat(F,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = groups3860910324918113789at_nat(F,A4) ) ) ) ).

% sum_diff1_nat
tff(fact_6058_sum__diff1__nat,axiom,
    ! [A: set_nat,A4: set_set_nat,F: fun_set_nat_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A4,insert_set_nat(A,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A4)),aa_set_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A4,insert_set_nat(A,bot_bot_set_set_nat))) = groups8294997508430121362at_nat(F,A4) ) ) ) ).

% sum_diff1_nat
tff(fact_6059_sum__diff1__nat,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A4))
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A4)),aa_Pro4243173770044397973at_nat(F,A)) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A4))
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = groups977919841031483927at_nat(F,A4) ) ) ) ).

% sum_diff1_nat
tff(fact_6060_sum__diff1__nat,axiom,
    ! [A: real,A4: set_real,F: fun_real_nat] :
      ( ( pp(aa_set_real_bool(member_real(A),A4))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),aa_real_nat(F,A)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A),A4))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups1935376822645274424al_nat(F,A4) ) ) ) ).

% sum_diff1_nat
tff(fact_6061_sum__diff1__nat,axiom,
    ! [A: int,A4: set_int,F: fun_int_nat] :
      ( ( pp(aa_set_int_bool(member_int(A),A4))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),aa_int_nat(F,A)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A),A4))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A4) ) ) ) ).

% sum_diff1_nat
tff(fact_6062_sum__diff1__nat,axiom,
    ! [A: nat,A4: set_nat,F: fun_nat_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A),A4))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A),A4))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) ) ) ) ).

% sum_diff1_nat
tff(fact_6063_Collect__conv__if,axiom,
    ! [P2: fun_Pr1607443439640623155t_bool,A: produc3843707927480180839at_nat] :
      ( ( pp(aa_Pro1809015098595223084t_bool(P2,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_jx(P2),A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) )
      & ( ~ pp(aa_Pro1809015098595223084t_bool(P2,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_jx(P2),A)) = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
tff(fact_6064_Collect__conv__if,axiom,
    ! [P2: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P2,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jy(P2),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P2,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jy(P2),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
tff(fact_6065_Collect__conv__if,axiom,
    ! [P2: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P2,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jz(P2),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P2,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jz(P2),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_6066_Collect__conv__if,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,A: product_prod_nat_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P2,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_ka(P2),A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P2,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_ka(P2),A)) = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
tff(fact_6067_Collect__conv__if,axiom,
    ! [P2: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P2,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_kb(P2),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P2,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_kb(P2),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if
tff(fact_6068_Collect__conv__if,axiom,
    ! [P2: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P2,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kc(P2),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P2,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kc(P2),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_6069_Collect__conv__if,axiom,
    ! [P2: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P2,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_kd(P2),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P2,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_kd(P2),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if
tff(fact_6070_Collect__conv__if2,axiom,
    ! [P2: fun_Pr1607443439640623155t_bool,A: produc3843707927480180839at_nat] :
      ( ( pp(aa_Pro1809015098595223084t_bool(P2,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_ke(P2),A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) )
      & ( ~ pp(aa_Pro1809015098595223084t_bool(P2,A))
       => ( collec6321179662152712658at_nat(aa_Pro2859564254541897054t_bool(aTP_Lamm_ke(P2),A)) = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6071_Collect__conv__if2,axiom,
    ! [P2: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P2,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_kf(P2),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P2,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_kf(P2),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6072_Collect__conv__if2,axiom,
    ! [P2: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P2,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_kg(P2),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P2,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_kg(P2),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6073_Collect__conv__if2,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,A: product_prod_nat_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P2,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_kh(P2),A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P2,A))
       => ( collec3392354462482085612at_nat(aa_Pro595088089689268830t_bool(aTP_Lamm_kh(P2),A)) = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6074_Collect__conv__if2,axiom,
    ! [P2: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P2,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ki(P2),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P2,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ki(P2),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
tff(fact_6075_Collect__conv__if2,axiom,
    ! [P2: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P2,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kj(P2),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P2,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kj(P2),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6076_Collect__conv__if2,axiom,
    ! [P2: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P2,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_kk(P2),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P2,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_kk(P2),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
tff(fact_6077_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A4))
           => ( 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),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_6078_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A4))
           => ( 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),A4) = aa_set_nat_real(groups6591440286371151544t_real(G),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_6079_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_mono
tff(fact_6080_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_mono
tff(fact_6081_le__of__int__ceiling,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ).

% le_of_int_ceiling
tff(fact_6082_le__of__int__ceiling,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ).

% le_of_int_ceiling
tff(fact_6083_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_6084_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_6085_infinite__finite__induct,axiom,
    ! [P2: fun_se8353504738346895699t_bool,A4: set_Pr4329608150637261639at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat] :
          ( ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A5))
         => pp(aa_set4148026973699692364t_bool(P2,A5)) )
     => ( pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat))
       => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
             => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X4),F4))
               => ( pp(aa_set4148026973699692364t_bool(P2,F4))
                 => pp(aa_set4148026973699692364t_bool(P2,insert9069300056098147895at_nat(X4,F4))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P2,A4)) ) ) ) ).

% infinite_finite_induct
tff(fact_6086_infinite__finite__induct,axiom,
    ! [P2: fun_set_set_nat_bool,A4: set_set_nat] :
      ( ! [A5: set_set_nat] :
          ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A5))
         => pp(aa_set_set_nat_bool(P2,A5)) )
     => ( pp(aa_set_set_nat_bool(P2,bot_bot_set_set_nat))
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
               => ( pp(aa_set_set_nat_bool(P2,F4))
                 => pp(aa_set_set_nat_bool(P2,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P2,A4)) ) ) ) ).

% infinite_finite_induct
tff(fact_6087_infinite__finite__induct,axiom,
    ! [P2: fun_set_complex_bool,A4: set_complex] :
      ( ! [A5: set_complex] :
          ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
         => pp(aa_set_complex_bool(P2,A5)) )
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X4: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
               => ( pp(aa_set_complex_bool(P2,F4))
                 => pp(aa_set_complex_bool(P2,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P2,A4)) ) ) ) ).

% infinite_finite_induct
tff(fact_6088_infinite__finite__induct,axiom,
    ! [P2: fun_se8074476271188067385t_bool,A4: set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat] :
          ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
         => pp(aa_set3135850010578026034t_bool(P2,A5)) )
     => ( pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat))
       => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
             => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),F4))
               => ( pp(aa_set3135850010578026034t_bool(P2,F4))
                 => pp(aa_set3135850010578026034t_bool(P2,insert8211810215607154385at_nat(X4,F4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P2,A4)) ) ) ) ).

% infinite_finite_induct
tff(fact_6089_infinite__finite__induct,axiom,
    ! [P2: fun_set_real_bool,A4: set_real] :
      ( ! [A5: set_real] :
          ( ~ pp(aa_set_real_bool(finite_finite_real,A5))
         => pp(aa_set_real_bool(P2,A5)) )
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [X4: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
               => ( pp(aa_set_real_bool(P2,F4))
                 => pp(aa_set_real_bool(P2,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P2,A4)) ) ) ) ).

% infinite_finite_induct
tff(fact_6090_infinite__finite__induct,axiom,
    ! [P2: fun_set_nat_bool,A4: set_nat] :
      ( ! [A5: set_nat] :
          ( ~ pp(aa_set_nat_bool(finite_finite_nat,A5))
         => pp(aa_set_nat_bool(P2,A5)) )
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [X4: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
               => ( pp(aa_set_nat_bool(P2,F4))
                 => pp(aa_set_nat_bool(P2,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P2,A4)) ) ) ) ).

% infinite_finite_induct
tff(fact_6091_infinite__finite__induct,axiom,
    ! [P2: fun_set_int_bool,A4: set_int] :
      ( ! [A5: set_int] :
          ( ~ pp(aa_set_int_bool(finite_finite_int,A5))
         => pp(aa_set_int_bool(P2,A5)) )
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [X4: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
               => ( pp(aa_set_int_bool(P2,F4))
                 => pp(aa_set_int_bool(P2,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P2,A4)) ) ) ) ).

% infinite_finite_induct
tff(fact_6092_finite__ne__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P2: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( ( F3 != bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(P2,insert9069300056098147895at_nat(X4,bot_bo228742789529271731at_nat)))
         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
               => ( ( F4 != bot_bo228742789529271731at_nat )
                 => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X4),F4))
                   => ( pp(aa_set4148026973699692364t_bool(P2,F4))
                     => pp(aa_set4148026973699692364t_bool(P2,insert9069300056098147895at_nat(X4,F4))) ) ) ) )
           => pp(aa_set4148026973699692364t_bool(P2,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6093_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P2: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X4: set_nat] : pp(aa_set_set_nat_bool(P2,insert_set_nat(X4,bot_bot_set_set_nat)))
         => ( ! [X4: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
                   => ( pp(aa_set_set_nat_bool(P2,F4))
                     => pp(aa_set_set_nat_bool(P2,insert_set_nat(X4,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P2,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6094_finite__ne__induct,axiom,
    ! [F3: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X4: complex] : pp(aa_set_complex_bool(P2,insert_complex(X4,bot_bot_set_complex)))
         => ( ! [X4: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
                   => ( pp(aa_set_complex_bool(P2,F4))
                     => pp(aa_set_complex_bool(P2,insert_complex(X4,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P2,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6095_finite__ne__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( ( F3 != bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(P2,insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))
         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( ( F4 != bot_bo2099793752762293965at_nat )
                 => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),F4))
                   => ( pp(aa_set3135850010578026034t_bool(P2,F4))
                     => pp(aa_set3135850010578026034t_bool(P2,insert8211810215607154385at_nat(X4,F4))) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P2,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6096_finite__ne__induct,axiom,
    ! [F3: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X4: real] : pp(aa_set_real_bool(P2,insert_real(X4,bot_bot_set_real)))
         => ( ! [X4: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
                   => ( pp(aa_set_real_bool(P2,F4))
                     => pp(aa_set_real_bool(P2,insert_real(X4,F4))) ) ) ) )
           => pp(aa_set_real_bool(P2,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6097_finite__ne__induct,axiom,
    ! [F3: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X4: nat] : pp(aa_set_nat_bool(P2,insert_nat(X4,bot_bot_set_nat)))
         => ( ! [X4: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
                   => ( pp(aa_set_nat_bool(P2,F4))
                     => pp(aa_set_nat_bool(P2,insert_nat(X4,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P2,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6098_finite__ne__induct,axiom,
    ! [F3: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X4: int] : pp(aa_set_int_bool(P2,insert_int(X4,bot_bot_set_int)))
         => ( ! [X4: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
                   => ( pp(aa_set_int_bool(P2,F4))
                     => pp(aa_set_int_bool(P2,insert_int(X4,F4))) ) ) ) )
           => pp(aa_set_int_bool(P2,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6099_finite__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P2: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat))
       => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
             => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X4),F4))
               => ( pp(aa_set4148026973699692364t_bool(P2,F4))
                 => pp(aa_set4148026973699692364t_bool(P2,insert9069300056098147895at_nat(X4,F4))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P2,F3)) ) ) ) ).

% finite_induct
tff(fact_6100_finite__induct,axiom,
    ! [F3: set_set_nat,P2: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(P2,bot_bot_set_set_nat))
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X4),F4))
               => ( pp(aa_set_set_nat_bool(P2,F4))
                 => pp(aa_set_set_nat_bool(P2,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P2,F3)) ) ) ) ).

% finite_induct
tff(fact_6101_finite__induct,axiom,
    ! [F3: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X4: complex,F4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
             => ( ~ pp(aa_set_complex_bool(member_complex(X4),F4))
               => ( pp(aa_set_complex_bool(P2,F4))
                 => pp(aa_set_complex_bool(P2,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P2,F3)) ) ) ) ).

% finite_induct
tff(fact_6102_finite__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat))
       => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
             => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),F4))
               => ( pp(aa_set3135850010578026034t_bool(P2,F4))
                 => pp(aa_set3135850010578026034t_bool(P2,insert8211810215607154385at_nat(X4,F4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P2,F3)) ) ) ) ).

% finite_induct
tff(fact_6103_finite__induct,axiom,
    ! [F3: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [X4: real,F4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F4))
             => ( ~ pp(aa_set_real_bool(member_real(X4),F4))
               => ( pp(aa_set_real_bool(P2,F4))
                 => pp(aa_set_real_bool(P2,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P2,F3)) ) ) ) ).

% finite_induct
tff(fact_6104_finite__induct,axiom,
    ! [F3: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [X4: nat,F4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F4))
             => ( ~ pp(aa_set_nat_bool(member_nat(X4),F4))
               => ( pp(aa_set_nat_bool(P2,F4))
                 => pp(aa_set_nat_bool(P2,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P2,F3)) ) ) ) ).

% finite_induct
tff(fact_6105_finite__induct,axiom,
    ! [F3: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [X4: int,F4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F4))
             => ( ~ pp(aa_set_int_bool(member_int(X4),F4))
               => ( pp(aa_set_int_bool(P2,F4))
                 => pp(aa_set_int_bool(P2,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P2,F3)) ) ) ) ).

% finite_induct
tff(fact_6106_finite_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A))
    <=> ( ( A = bot_bo228742789529271731at_nat )
        | ? [A7: set_Pr4329608150637261639at_nat,A6: produc3843707927480180839at_nat] :
            ( ( A = insert9069300056098147895at_nat(A6,A7) )
            & pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_6107_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
    <=> ( ( A = bot_bot_set_complex )
        | ? [A7: set_complex,A6: complex] :
            ( ( A = insert_complex(A6,A7) )
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,A7)) ) ) ) ).

% finite.simps
tff(fact_6108_finite_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A))
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        | ? [A7: set_Pr1261947904930325089at_nat,A6: product_prod_nat_nat] :
            ( ( A = insert8211810215607154385at_nat(A6,A7) )
            & pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_6109_finite_Osimps,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
    <=> ( ( A = bot_bot_set_real )
        | ? [A7: set_real,A6: real] :
            ( ( A = insert_real(A6,A7) )
            & pp(aa_set_real_bool(finite_finite_real,A7)) ) ) ) ).

% finite.simps
tff(fact_6110_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A6: nat] :
            ( ( A = insert_nat(A6,A7) )
            & pp(aa_set_nat_bool(finite_finite_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_6111_finite_Osimps,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A6: int] :
            ( ( A = insert_int(A6,A7) )
            & pp(aa_set_int_bool(finite_finite_int,A7)) ) ) ) ).

% finite.simps
tff(fact_6112_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A))
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A5: set_Pr4329608150637261639at_nat] :
              ( ? [A3: produc3843707927480180839at_nat] : A = insert9069300056098147895at_nat(A3,A5)
             => ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A5)) ) ) ) ).

% finite.cases
tff(fact_6113_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A5: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A5)
             => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A5)) ) ) ) ).

% finite.cases
tff(fact_6114_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A))
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A5: set_Pr1261947904930325089at_nat] :
              ( ? [A3: product_prod_nat_nat] : A = insert8211810215607154385at_nat(A3,A5)
             => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5)) ) ) ) ).

% finite.cases
tff(fact_6115_finite_Ocases,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( A != bot_bot_set_real )
       => ~ ! [A5: set_real] :
              ( ? [A3: real] : A = insert_real(A3,A5)
             => ~ pp(aa_set_real_bool(finite_finite_real,A5)) ) ) ) ).

% finite.cases
tff(fact_6116_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A5: set_nat] :
              ( ? [A3: nat] : A = insert_nat(A3,A5)
             => ~ pp(aa_set_nat_bool(finite_finite_nat,A5)) ) ) ) ).

% finite.cases
tff(fact_6117_finite_Ocases,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( A != bot_bot_set_int )
       => ~ ! [A5: set_int] :
              ( ? [A3: int] : A = insert_int(A3,A5)
             => ~ pp(aa_set_int_bool(finite_finite_int,A5)) ) ) ) ).

% finite.cases
tff(fact_6118_subset__singleton__iff,axiom,
    ! [X6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,X6),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))
    <=> ( ( X6 = bot_bo228742789529271731at_nat )
        | ( X6 = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_6119_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X6),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))
    <=> ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6 = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_6120_subset__singleton__iff,axiom,
    ! [X6: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),insert_real(A,bot_bot_set_real)))
    <=> ( ( X6 = bot_bot_set_real )
        | ( X6 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_6121_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),insert_nat(A,bot_bot_set_nat)))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_6122_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),insert_int(A,bot_bot_set_int)))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_6123_subset__singletonD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)))
     => ( ( A4 = bot_bo228742789529271731at_nat )
        | ( A4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singletonD
tff(fact_6124_subset__singletonD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))
     => ( ( A4 = bot_bo2099793752762293965at_nat )
        | ( A4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_6125_subset__singletonD,axiom,
    ! [A4: set_real,X: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),insert_real(X,bot_bot_set_real)))
     => ( ( A4 = bot_bot_set_real )
        | ( A4 = insert_real(X,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_6126_subset__singletonD,axiom,
    ! [A4: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),insert_nat(X,bot_bot_set_nat)))
     => ( ( A4 = bot_bot_set_nat )
        | ( A4 = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_6127_subset__singletonD,axiom,
    ! [A4: set_int,X: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),insert_int(X,bot_bot_set_int)))
     => ( ( A4 = bot_bot_set_int )
        | ( A4 = insert_int(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_6128_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_6129_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_6130_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( set_or1222579329274155063t_real(A,B) = insert_real(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_6131_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A4))
     => ( minus_3314409938677909166at_nat(insert9069300056098147895at_nat(X,A4),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_6132_Diff__insert__absorb,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A4))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A4),insert_set_nat(X,bot_bot_set_set_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_6133_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A4))
     => ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_6134_Diff__insert__absorb,axiom,
    ! [X: real,A4: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),A4))
     => ( minus_minus_set_real(insert_real(X,A4),insert_real(X,bot_bot_set_real)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_6135_Diff__insert__absorb,axiom,
    ! [X: int,A4: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),A4))
     => ( minus_minus_set_int(insert_int(X,A4),insert_int(X,bot_bot_set_int)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_6136_Diff__insert__absorb,axiom,
    ! [X: nat,A4: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
     => ( minus_minus_set_nat(insert_nat(X,A4),insert_nat(X,bot_bot_set_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_6137_Diff__insert2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,B4)) = minus_3314409938677909166at_nat(minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)),B4) ).

% Diff_insert2
tff(fact_6138_Diff__insert2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,B4)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)),B4) ).

% Diff_insert2
tff(fact_6139_Diff__insert2,axiom,
    ! [A4: set_real,A: real,B4: set_real] : minus_minus_set_real(A4,insert_real(A,B4)) = minus_minus_set_real(minus_minus_set_real(A4,insert_real(A,bot_bot_set_real)),B4) ).

% Diff_insert2
tff(fact_6140_Diff__insert2,axiom,
    ! [A4: set_int,A: int,B4: set_int] : minus_minus_set_int(A4,insert_int(A,B4)) = minus_minus_set_int(minus_minus_set_int(A4,insert_int(A,bot_bot_set_int)),B4) ).

% Diff_insert2
tff(fact_6141_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : minus_minus_set_nat(A4,insert_nat(A,B4)) = minus_minus_set_nat(minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat)),B4) ).

% Diff_insert2
tff(fact_6142_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A),A4))
     => ( insert9069300056098147895at_nat(A,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_6143_insert__Diff,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A4,insert_set_nat(A,bot_bot_set_set_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_6144_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A4))
     => ( insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_6145_insert__Diff,axiom,
    ! [A: real,A4: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A4))
     => ( insert_real(A,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = A4 ) ) ).

% insert_Diff
tff(fact_6146_insert__Diff,axiom,
    ! [A: int,A4: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A4))
     => ( insert_int(A,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = A4 ) ) ).

% insert_Diff
tff(fact_6147_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A4))
     => ( insert_nat(A,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_6148_Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(A,B4)) = minus_3314409938677909166at_nat(minus_3314409938677909166at_nat(A4,B4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% Diff_insert
tff(fact_6149_Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(A,B4)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A4,B4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% Diff_insert
tff(fact_6150_Diff__insert,axiom,
    ! [A4: set_real,A: real,B4: set_real] : minus_minus_set_real(A4,insert_real(A,B4)) = minus_minus_set_real(minus_minus_set_real(A4,B4),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_6151_Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] : minus_minus_set_int(A4,insert_int(A,B4)) = minus_minus_set_int(minus_minus_set_int(A4,B4),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_6152_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : minus_minus_set_nat(A4,insert_nat(A,B4)) = minus_minus_set_nat(minus_minus_set_nat(A4,B4),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_6153_sum__subtractf__nat,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X4)),aa_Pro4243173770044397973at_nat(F,X4))) )
     => ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_kl(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A4)),groups977919841031483927at_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6154_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_km(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6155_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G,X4)),aa_set_nat_nat(F,X4))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kn(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A4)),groups8294997508430121362at_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6156_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ko(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6157_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_kp(G),F)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6158_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6159_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M2: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6160_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).

% not_mask_negative_int
tff(fact_6161_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M2: nat,K2: nat,N: 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,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(G),K2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6162_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M2: nat,K2: nat,N: 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,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_kt(G),K2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6163_sum__eq__Suc0__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups4541462559716669496nt_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6164_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups5693394587270226106ex_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6165_sum__eq__Suc0__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( ( groups977919841031483927at_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
            & ( aa_Pro4243173770044397973at_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6166_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6167_sum__SucD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,N) )
     => ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) ) ) ).

% sum_SucD
tff(fact_6168_sum__eq__1__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups4541462559716669496nt_nat(F,A4) = one_one_nat )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_nat(F,X5) = one_one_nat )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6169_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups5693394587270226106ex_nat(F,A4) = one_one_nat )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_nat(F,X5) = one_one_nat )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6170_sum__eq__1__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( ( groups977919841031483927at_nat(F,A4) = one_one_nat )
      <=> ? [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
            & ( aa_Pro4243173770044397973at_nat(F,X5) = one_one_nat )
            & ! [Xa3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6171_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = one_one_nat )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_nat(F,X5) = one_one_nat )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A4))
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6172_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A)) ) ).

% ceiling_le
tff(fact_6173_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),A)) ) ).

% ceiling_le
tff(fact_6174_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% ceiling_le_iff
tff(fact_6175_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% ceiling_le_iff
tff(fact_6176_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_6177_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_6178_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))) ).

% ceiling_add_le
tff(fact_6179_ceiling__add__le,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))) ).

% ceiling_add_le
tff(fact_6180_finite__ranking__induct,axiom,
    ! [S2: set_complex,P2: fun_set_complex_bool,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X4: complex,S6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S6))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X4))) )
               => ( pp(aa_set_complex_bool(P2,S6))
                 => pp(aa_set_complex_bool(P2,insert_complex(X4,S6))) ) ) )
         => pp(aa_set_complex_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6181_finite__ranking__induct,axiom,
    ! [S2: set_real,P2: fun_set_real_bool,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [X4: real,S6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S6))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X4))) )
               => ( pp(aa_set_real_bool(P2,S6))
                 => pp(aa_set_real_bool(P2,insert_real(X4,S6))) ) ) )
         => pp(aa_set_real_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6182_finite__ranking__induct,axiom,
    ! [S2: set_nat,P2: fun_set_nat_bool,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [X4: nat,S6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S6))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X4))) )
               => ( pp(aa_set_nat_bool(P2,S6))
                 => pp(aa_set_nat_bool(P2,insert_nat(X4,S6))) ) ) )
         => pp(aa_set_nat_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6183_finite__ranking__induct,axiom,
    ! [S2: set_int,P2: fun_set_int_bool,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [X4: int,S6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S6))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X4))) )
               => ( pp(aa_set_int_bool(P2,S6))
                 => pp(aa_set_int_bool(P2,insert_int(X4,S6))) ) ) )
         => pp(aa_set_int_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6184_finite__ranking__induct,axiom,
    ! [S2: set_complex,P2: fun_set_complex_bool,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X4: complex,S6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S6))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X4))) )
               => ( pp(aa_set_complex_bool(P2,S6))
                 => pp(aa_set_complex_bool(P2,insert_complex(X4,S6))) ) ) )
         => pp(aa_set_complex_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6185_finite__ranking__induct,axiom,
    ! [S2: set_real,P2: fun_set_real_bool,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [X4: real,S6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S6))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X4))) )
               => ( pp(aa_set_real_bool(P2,S6))
                 => pp(aa_set_real_bool(P2,insert_real(X4,S6))) ) ) )
         => pp(aa_set_real_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6186_finite__ranking__induct,axiom,
    ! [S2: set_nat,P2: fun_set_nat_bool,F: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [X4: nat,S6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S6))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,Y4)),aa_nat_num(F,X4))) )
               => ( pp(aa_set_nat_bool(P2,S6))
                 => pp(aa_set_nat_bool(P2,insert_nat(X4,S6))) ) ) )
         => pp(aa_set_nat_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6187_finite__ranking__induct,axiom,
    ! [S2: set_int,P2: fun_set_int_bool,F: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [X4: int,S6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S6))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,Y4)),aa_int_num(F,X4))) )
               => ( pp(aa_set_int_bool(P2,S6))
                 => pp(aa_set_int_bool(P2,insert_int(X4,S6))) ) ) )
         => pp(aa_set_int_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6188_finite__ranking__induct,axiom,
    ! [S2: set_complex,P2: fun_set_complex_bool,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X4: complex,S6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S6))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,Y4)),aa_complex_nat(F,X4))) )
               => ( pp(aa_set_complex_bool(P2,S6))
                 => pp(aa_set_complex_bool(P2,insert_complex(X4,S6))) ) ) )
         => pp(aa_set_complex_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6189_finite__ranking__induct,axiom,
    ! [S2: set_real,P2: fun_set_real_bool,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [X4: real,S6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S6))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,Y4)),aa_real_nat(F,X4))) )
               => ( pp(aa_set_real_bool(P2,S6))
                 => pp(aa_set_real_bool(P2,insert_real(X4,S6))) ) ) )
         => pp(aa_set_real_bool(P2,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6190_finite__linorder__max__induct,axiom,
    ! [A4: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [B3: real,A5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A5))
             => ( ! [X3: real] :
                    ( pp(aa_set_real_bool(member_real(X3),A5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B3)) )
               => ( pp(aa_set_real_bool(P2,A5))
                 => pp(aa_set_real_bool(P2,insert_real(B3,A5))) ) ) )
         => pp(aa_set_real_bool(P2,A4)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6191_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,P2: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A4))
     => ( pp(aa_set_rat_bool(P2,bot_bot_set_rat))
       => ( ! [B3: rat,A5: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A5))
             => ( ! [X3: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X3),A5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),B3)) )
               => ( pp(aa_set_rat_bool(P2,A5))
                 => pp(aa_set_rat_bool(P2,insert_rat(B3,A5))) ) ) )
         => pp(aa_set_rat_bool(P2,A4)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6192_finite__linorder__max__induct,axiom,
    ! [A4: set_num,P2: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A4))
     => ( pp(aa_set_num_bool(P2,bot_bot_set_num))
       => ( ! [B3: num,A5: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A5))
             => ( ! [X3: num] :
                    ( pp(aa_set_num_bool(member_num(X3),A5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),B3)) )
               => ( pp(aa_set_num_bool(P2,A5))
                 => pp(aa_set_num_bool(P2,insert_num(B3,A5))) ) ) )
         => pp(aa_set_num_bool(P2,A4)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6193_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [B3: nat,A5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A5))
             => ( ! [X3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X3),A5))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),B3)) )
               => ( pp(aa_set_nat_bool(P2,A5))
                 => pp(aa_set_nat_bool(P2,insert_nat(B3,A5))) ) ) )
         => pp(aa_set_nat_bool(P2,A4)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6194_finite__linorder__max__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [B3: int,A5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A5))
             => ( ! [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),A5))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),B3)) )
               => ( pp(aa_set_int_bool(P2,A5))
                 => pp(aa_set_int_bool(P2,insert_int(B3,A5))) ) ) )
         => pp(aa_set_int_bool(P2,A4)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6195_finite__linorder__min__induct,axiom,
    ! [A4: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [B3: real,A5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A5))
             => ( ! [X3: real] :
                    ( pp(aa_set_real_bool(member_real(X3),A5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X3)) )
               => ( pp(aa_set_real_bool(P2,A5))
                 => pp(aa_set_real_bool(P2,insert_real(B3,A5))) ) ) )
         => pp(aa_set_real_bool(P2,A4)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6196_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,P2: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A4))
     => ( pp(aa_set_rat_bool(P2,bot_bot_set_rat))
       => ( ! [B3: rat,A5: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A5))
             => ( ! [X3: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X3),A5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),X3)) )
               => ( pp(aa_set_rat_bool(P2,A5))
                 => pp(aa_set_rat_bool(P2,insert_rat(B3,A5))) ) ) )
         => pp(aa_set_rat_bool(P2,A4)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6197_finite__linorder__min__induct,axiom,
    ! [A4: set_num,P2: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A4))
     => ( pp(aa_set_num_bool(P2,bot_bot_set_num))
       => ( ! [B3: num,A5: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A5))
             => ( ! [X3: num] :
                    ( pp(aa_set_num_bool(member_num(X3),A5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),X3)) )
               => ( pp(aa_set_num_bool(P2,A5))
                 => pp(aa_set_num_bool(P2,insert_num(B3,A5))) ) ) )
         => pp(aa_set_num_bool(P2,A4)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6198_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [B3: nat,A5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A5))
             => ( ! [X3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X3),A5))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),X3)) )
               => ( pp(aa_set_nat_bool(P2,A5))
                 => pp(aa_set_nat_bool(P2,insert_nat(B3,A5))) ) ) )
         => pp(aa_set_nat_bool(P2,A4)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6199_finite__linorder__min__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [B3: int,A5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A5))
             => ( ! [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),A5))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),X3)) )
               => ( pp(aa_set_int_bool(P2,A5))
                 => pp(aa_set_int_bool(P2,insert_int(B3,A5))) ) ) )
         => pp(aa_set_int_bool(P2,A4)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6200_sum_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A4)) = groups1935376822645274424al_nat(G,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6201_sum_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( groups4541462559716669496nt_nat(G,insert_int(X,A4)) = groups4541462559716669496nt_nat(G,A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( groups4541462559716669496nt_nat(G,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6202_sum_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups5693394587270226106ex_nat(G,insert_complex(X,A4)) = groups5693394587270226106ex_nat(G,A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups5693394587270226106ex_nat(G,insert_complex(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6203_sum_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1932886352136224148al_int(G,insert_real(X,A4)) = groups1932886352136224148al_int(G,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1932886352136224148al_int(G,insert_real(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6204_sum_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(X,A4)) = aa_set_nat_int(groups3539618377306564664at_int(G),A4) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_int(groups3539618377306564664at_int(G),insert_nat(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,X)),aa_set_nat_int(groups3539618377306564664at_int(G),A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6205_sum_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups5690904116761175830ex_int(G,insert_complex(X,A4)) = groups5690904116761175830ex_int(G,A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups5690904116761175830ex_int(G,insert_complex(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6206_sum_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups8097168146408367636l_real(G,insert_real(X,A4)) = groups8097168146408367636l_real(G,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups8097168146408367636l_real(G,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6207_sum_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( groups8778361861064173332t_real(G,insert_int(X,A4)) = groups8778361861064173332t_real(G,A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( groups8778361861064173332t_real(G,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6208_sum_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups5808333547571424918x_real(G,insert_complex(X,A4)) = groups5808333547571424918x_real(G,A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups5808333547571424918x_real(G,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6209_sum_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1300246762558778688al_rat(G,insert_real(X,A4)) = groups1300246762558778688al_rat(G,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1300246762558778688al_rat(G,insert_real(X,A4)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,A4)) ) ) ) ) ).

% sum.insert_if
tff(fact_6210_finite__subset__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,F3),A4))
       => ( pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat))
         => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
               => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),A4))
                 => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),F4))
                   => ( pp(aa_set4148026973699692364t_bool(P2,F4))
                     => pp(aa_set4148026973699692364t_bool(P2,insert9069300056098147895at_nat(A3,F4))) ) ) ) )
           => pp(aa_set4148026973699692364t_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6211_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A4: set_set_nat,P2: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A4))
       => ( pp(aa_set_set_nat_bool(P2,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A4))
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                   => ( pp(aa_set_set_nat_bool(P2,F4))
                     => pp(aa_set_set_nat_bool(P2,insert_set_nat(A3,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6212_finite__subset__induct,axiom,
    ! [F3: set_complex,A4: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A4))
       => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A4))
                 => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                   => ( pp(aa_set_complex_bool(P2,F4))
                     => pp(aa_set_complex_bool(P2,insert_complex(A3,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6213_finite__subset__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,F3),A4))
       => ( pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat))
         => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),A4))
                 => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),F4))
                   => ( pp(aa_set3135850010578026034t_bool(P2,F4))
                     => pp(aa_set3135850010578026034t_bool(P2,insert8211810215607154385at_nat(A3,F4))) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6214_finite__subset__induct,axiom,
    ! [F3: set_real,A4: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A4))
       => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
         => ( ! [A3: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A3),A4))
                 => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                   => ( pp(aa_set_real_bool(P2,F4))
                     => pp(aa_set_real_bool(P2,insert_real(A3,F4))) ) ) ) )
           => pp(aa_set_real_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6215_finite__subset__induct,axiom,
    ! [F3: set_nat,A4: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F3),A4))
       => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
         => ( ! [A3: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A3),A4))
                 => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                   => ( pp(aa_set_nat_bool(P2,F4))
                     => pp(aa_set_nat_bool(P2,insert_nat(A3,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6216_finite__subset__induct,axiom,
    ! [F3: set_int,A4: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A4))
       => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
         => ( ! [A3: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A3),A4))
                 => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                   => ( pp(aa_set_int_bool(P2,F4))
                     => pp(aa_set_int_bool(P2,insert_int(A3,F4))) ) ) ) )
           => pp(aa_set_int_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6217_finite__subset__induct_H,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F3))
     => ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,F3),A4))
       => ( pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat))
         => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,F4))
               => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),A4))
                 => ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,F4),A4))
                   => ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),F4))
                     => ( pp(aa_set4148026973699692364t_bool(P2,F4))
                       => pp(aa_set4148026973699692364t_bool(P2,insert9069300056098147895at_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set4148026973699692364t_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6218_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A4: set_set_nat,P2: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A4))
       => ( pp(aa_set_set_nat_bool(P2,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F4: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A4))
                 => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F4),A4))
                   => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                     => ( pp(aa_set_set_nat_bool(P2,F4))
                       => pp(aa_set_set_nat_bool(P2,insert_set_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_set_nat_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6219_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A4: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A4))
       => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A4))
                 => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F4),A4))
                   => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                     => ( pp(aa_set_complex_bool(P2,F4))
                       => pp(aa_set_complex_bool(P2,insert_complex(A3,F4))) ) ) ) ) )
           => pp(aa_set_complex_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6220_finite__subset__induct_H,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,F3),A4))
       => ( pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat))
         => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F4))
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),A4))
                 => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,F4),A4))
                   => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),F4))
                     => ( pp(aa_set3135850010578026034t_bool(P2,F4))
                       => pp(aa_set3135850010578026034t_bool(P2,insert8211810215607154385at_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set3135850010578026034t_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6221_finite__subset__induct_H,axiom,
    ! [F3: set_real,A4: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A4))
       => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
         => ( ! [A3: real,F4: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F4))
               => ( pp(aa_set_real_bool(member_real(A3),A4))
                 => ( pp(aa_set_real_bool(ord_less_eq_set_real(F4),A4))
                   => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                     => ( pp(aa_set_real_bool(P2,F4))
                       => pp(aa_set_real_bool(P2,insert_real(A3,F4))) ) ) ) ) )
           => pp(aa_set_real_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6222_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A4: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F3),A4))
       => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
         => ( ! [A3: nat,F4: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F4))
               => ( pp(aa_set_nat_bool(member_nat(A3),A4))
                 => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F4),A4))
                   => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                     => ( pp(aa_set_nat_bool(P2,F4))
                       => pp(aa_set_nat_bool(P2,insert_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_nat_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6223_finite__subset__induct_H,axiom,
    ! [F3: set_int,A4: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A4))
       => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
         => ( ! [A3: int,F4: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F4))
               => ( pp(aa_set_int_bool(member_int(A3),A4))
                 => ( pp(aa_set_int_bool(ord_less_eq_set_int(F4),A4))
                   => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                     => ( pp(aa_set_int_bool(P2,F4))
                       => pp(aa_set_int_bool(P2,insert_int(A3,F4))) ) ) ) ) )
           => pp(aa_set_int_bool(P2,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6224_finite__empty__induct,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,P2: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4))
     => ( pp(aa_set4148026973699692364t_bool(P2,A4))
       => ( ! [A3: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A5))
             => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(A3),A5))
               => ( pp(aa_set4148026973699692364t_bool(P2,A5))
                 => pp(aa_set4148026973699692364t_bool(P2,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(A3,bot_bo228742789529271731at_nat)))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6225_finite__empty__induct,axiom,
    ! [A4: set_set_nat,P2: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( pp(aa_set_set_nat_bool(P2,A4))
       => ( ! [A3: set_nat,A5: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A5))
             => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A5))
               => ( pp(aa_set_set_nat_bool(P2,A5))
                 => pp(aa_set_set_nat_bool(P2,minus_2163939370556025621et_nat(A5,insert_set_nat(A3,bot_bot_set_set_nat)))) ) ) )
         => pp(aa_set_set_nat_bool(P2,bot_bot_set_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6226_finite__empty__induct,axiom,
    ! [A4: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(P2,A4))
       => ( ! [A3: complex,A5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
             => ( pp(aa_set_complex_bool(member_complex(A3),A5))
               => ( pp(aa_set_complex_bool(P2,A5))
                 => pp(aa_set_complex_bool(P2,minus_811609699411566653omplex(A5,insert_complex(A3,bot_bot_set_complex)))) ) ) )
         => pp(aa_set_complex_bool(P2,bot_bot_set_complex)) ) ) ) ).

% finite_empty_induct
tff(fact_6227_finite__empty__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( pp(aa_set3135850010578026034t_bool(P2,A4))
       => ( ! [A3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A3),A5))
               => ( pp(aa_set3135850010578026034t_bool(P2,A5))
                 => pp(aa_set3135850010578026034t_bool(P2,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(A3,bot_bo2099793752762293965at_nat)))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6228_finite__empty__induct,axiom,
    ! [A4: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(P2,A4))
       => ( ! [A3: real,A5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A5))
             => ( pp(aa_set_real_bool(member_real(A3),A5))
               => ( pp(aa_set_real_bool(P2,A5))
                 => pp(aa_set_real_bool(P2,minus_minus_set_real(A5,insert_real(A3,bot_bot_set_real)))) ) ) )
         => pp(aa_set_real_bool(P2,bot_bot_set_real)) ) ) ) ).

% finite_empty_induct
tff(fact_6229_finite__empty__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(P2,A4))
       => ( ! [A3: int,A5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A5))
             => ( pp(aa_set_int_bool(member_int(A3),A5))
               => ( pp(aa_set_int_bool(P2,A5))
                 => pp(aa_set_int_bool(P2,minus_minus_set_int(A5,insert_int(A3,bot_bot_set_int)))) ) ) )
         => pp(aa_set_int_bool(P2,bot_bot_set_int)) ) ) ) ).

% finite_empty_induct
tff(fact_6230_finite__empty__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(P2,A4))
       => ( ! [A3: nat,A5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A5))
             => ( pp(aa_set_nat_bool(member_nat(A3),A5))
               => ( pp(aa_set_nat_bool(P2,A5))
                 => pp(aa_set_nat_bool(P2,minus_minus_set_nat(A5,insert_nat(A3,bot_bot_set_nat)))) ) ) )
         => pp(aa_set_nat_bool(P2,bot_bot_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6231_infinite__coinduct,axiom,
    ! [X6: fun_se8353504738346895699t_bool,A4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(X6,A4))
     => ( ! [A5: set_Pr4329608150637261639at_nat] :
            ( pp(aa_set4148026973699692364t_bool(X6,A5))
           => ? [X3: produc3843707927480180839at_nat] :
                ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X3),A5))
                & ( pp(aa_set4148026973699692364t_bool(X6,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat))))
                  | ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)))) ) ) )
       => ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A4)) ) ) ).

% infinite_coinduct
tff(fact_6232_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_bool,A4: set_complex] :
      ( pp(aa_set_complex_bool(X6,A4))
     => ( ! [A5: set_complex] :
            ( pp(aa_set_complex_bool(X6,A5))
           => ? [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),A5))
                & ( pp(aa_set_complex_bool(X6,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex))))
                  | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex)))) ) ) )
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ).

% infinite_coinduct
tff(fact_6233_infinite__coinduct,axiom,
    ! [X6: fun_se8074476271188067385t_bool,A4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(X6,A4))
     => ( ! [A5: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(X6,A5))
           => ? [X3: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A5))
                & ( pp(aa_set3135850010578026034t_bool(X6,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat))))
                  | ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)))) ) ) )
       => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4)) ) ) ).

% infinite_coinduct
tff(fact_6234_infinite__coinduct,axiom,
    ! [X6: fun_set_real_bool,A4: set_real] :
      ( pp(aa_set_real_bool(X6,A4))
     => ( ! [A5: set_real] :
            ( pp(aa_set_real_bool(X6,A5))
           => ? [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),A5))
                & ( pp(aa_set_real_bool(X6,minus_minus_set_real(A5,insert_real(X3,bot_bot_set_real))))
                  | ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A5,insert_real(X3,bot_bot_set_real)))) ) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,A4)) ) ) ).

% infinite_coinduct
tff(fact_6235_infinite__coinduct,axiom,
    ! [X6: fun_set_int_bool,A4: set_int] :
      ( pp(aa_set_int_bool(X6,A4))
     => ( ! [A5: set_int] :
            ( pp(aa_set_int_bool(X6,A5))
           => ? [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),A5))
                & ( pp(aa_set_int_bool(X6,minus_minus_set_int(A5,insert_int(X3,bot_bot_set_int))))
                  | ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A5,insert_int(X3,bot_bot_set_int)))) ) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ).

% infinite_coinduct
tff(fact_6236_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_bool,A4: set_nat] :
      ( pp(aa_set_nat_bool(X6,A4))
     => ( ! [A5: set_nat] :
            ( pp(aa_set_nat_bool(X6,A5))
           => ? [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),A5))
                & ( pp(aa_set_nat_bool(X6,minus_minus_set_nat(A5,insert_nat(X3,bot_bot_set_nat))))
                  | ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A5,insert_nat(X3,bot_bot_set_nat)))) ) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ).

% infinite_coinduct
tff(fact_6237_infinite__remove,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,S2))
     => ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,minus_3314409938677909166at_nat(S2,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)))) ) ).

% infinite_remove
tff(fact_6238_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ).

% infinite_remove
tff(fact_6239_infinite__remove,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
     => ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))) ) ).

% infinite_remove
tff(fact_6240_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S2))
     => ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ).

% infinite_remove
tff(fact_6241_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
     => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) ).

% infinite_remove
tff(fact_6242_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat)))) ) ).

% infinite_remove
tff(fact_6243_subset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),insert9069300056098147895at_nat(X,B4)))
    <=> ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A4))
         => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B4)) )
        & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A4))
         => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_6244_subset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),insert_set_nat(X,B4)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A4))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat))),B4)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A4))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_6245_subset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X,B4)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A4))
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B4)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A4))
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_6246_subset__insert__iff,axiom,
    ! [A4: set_real,X: real,B4: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),insert_real(X,B4)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),A4))
         => pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A4,insert_real(X,bot_bot_set_real))),B4)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_6247_subset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),insert_nat(X,B4)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A4,insert_nat(X,bot_bot_set_nat))),B4)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_6248_subset__insert__iff,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),insert_int(X,B4)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),A4))
         => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A4,insert_int(X,bot_bot_set_int))),B4)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_6249_Diff__single__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B4))
     => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),insert9069300056098147895at_nat(X,B4))) ) ).

% Diff_single_insert
tff(fact_6250_Diff__single__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B4))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X,B4))) ) ).

% Diff_single_insert
tff(fact_6251_Diff__single__insert,axiom,
    ! [A4: set_real,X: real,B4: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A4,insert_real(X,bot_bot_set_real))),B4))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A4),insert_real(X,B4))) ) ).

% Diff_single_insert
tff(fact_6252_Diff__single__insert,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A4,insert_nat(X,bot_bot_set_nat))),B4))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),insert_nat(X,B4))) ) ).

% Diff_single_insert
tff(fact_6253_Diff__single__insert,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A4,insert_int(X,bot_bot_set_int))),B4))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A4),insert_int(X,B4))) ) ).

% Diff_single_insert
tff(fact_6254_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_6255_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( insert_nat(M2,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_6256_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_6257_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N) = insert_nat(M2,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_6258_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I2: nat,X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I2,X))),insert8211810215607154385at_nat(X,set_Pr5648618587558075414at_nat(Xs)))) ).

% set_update_subset_insert
tff(fact_6259_set__update__subset__insert,axiom,
    ! [Xs: list_real,I2: nat,X: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I2,X))),insert_real(X,set_real2(Xs)))) ).

% set_update_subset_insert
tff(fact_6260_set__update__subset__insert,axiom,
    ! [Xs: list_P5464809261938338413at_nat,I2: nat,X: produc3843707927480180839at_nat] : pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,I2,X))),insert9069300056098147895at_nat(X,set_Pr3765526544606949372at_nat(Xs)))) ).

% set_update_subset_insert
tff(fact_6261_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I2: nat,X: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I2,X))),insert_nat(X,set_nat2(Xs)))) ).

% set_update_subset_insert
tff(fact_6262_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I2,X)),insert_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_6263_set__update__subset__insert,axiom,
    ! [Xs: list_int,I2: nat,X: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I2,X))),insert_int(X,set_int2(Xs)))) ).

% set_update_subset_insert
tff(fact_6264_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(insert9069300056098147895at_nat(X,A4)) = minus_3314409938677909166at_nat(uminus935396558254630718at_nat(A4),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% Compl_insert
tff(fact_6265_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(insert8211810215607154385at_nat(X,A4)) = minus_1356011639430497352at_nat(uminus6524753893492686040at_nat(A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% Compl_insert
tff(fact_6266_Compl__insert,axiom,
    ! [X: real,A4: set_real] : uminus612125837232591019t_real(insert_real(X,A4)) = minus_minus_set_real(uminus612125837232591019t_real(A4),insert_real(X,bot_bot_set_real)) ).

% Compl_insert
tff(fact_6267_Compl__insert,axiom,
    ! [X: int,A4: set_int] : uminus1532241313380277803et_int(insert_int(X,A4)) = minus_minus_set_int(uminus1532241313380277803et_int(A4),insert_int(X,bot_bot_set_int)) ).

% Compl_insert
tff(fact_6268_Compl__insert,axiom,
    ! [X: nat,A4: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A4)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A4),insert_nat(X,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_6269_sum__power__add,axiom,
    ! [X: complex,M2: nat,I6: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ku(X),M2),I6) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M2)),groups2073611262835488442omplex(power_power_complex(X),I6)) ).

% sum_power_add
tff(fact_6270_sum__power__add,axiom,
    ! [X: rat,M2: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_kv(X),M2),I6) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M2)),groups2906978787729119204at_rat(power_power_rat(X),I6)) ).

% sum_power_add
tff(fact_6271_sum__power__add,axiom,
    ! [X: int,M2: nat,I6: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_kw(X),M2)),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M2)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),I6)) ).

% sum_power_add
tff(fact_6272_sum__power__add,axiom,
    ! [X: real,M2: nat,I6: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_kx(X),M2)),I6) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),I6)) ).

% sum_power_add
tff(fact_6273_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ky(G),N),M2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) ).

% sum.atLeastAtMost_rev
tff(fact_6274_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_kz(G),N),M2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) ).

% sum.atLeastAtMost_rev
tff(fact_6275_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_6276_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_cb,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_da(N),C2))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_6277_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_cb,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_dc,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_6278_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real))) ).

% of_int_ceiling_le_add_one
tff(fact_6279_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(plus_plus_rat(R2),one_one_rat))) ).

% of_int_ceiling_le_add_one
tff(fact_6280_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_6281_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_6282_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_6283_sum__diff__nat,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B4),A4))
       => ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A4)),groups977919841031483927at_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_6284_sum__diff__nat,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_6285_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_6286_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K2: 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)),K2)) = groups2073611262835488442omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6287_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K2: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K2)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6288_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K2: 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)),K2)) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6289_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K2: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K2)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6290_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K2: 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)),K2)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6291_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: 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,N))) = 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),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6292_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6293_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: 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,N))) = 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),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6294_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: 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,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6295_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6296_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6297_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6298_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6299_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M2)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6300_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M2)),groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6301_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6302_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M2)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6303_remove__induct,axiom,
    ! [P2: fun_se8353504738346895699t_bool,B4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat))
     => ( ( ~ pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,B4))
         => pp(aa_set4148026973699692364t_bool(P2,B4)) )
       => ( ! [A5: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A5))
             => ( ( A5 != bot_bo228742789529271731at_nat )
               => ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A5),B4))
                 => ( ! [X3: produc3843707927480180839at_nat] :
                        ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X3),A5))
                       => pp(aa_set4148026973699692364t_bool(P2,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)))) )
                   => pp(aa_set4148026973699692364t_bool(P2,A5)) ) ) ) )
         => pp(aa_set4148026973699692364t_bool(P2,B4)) ) ) ) ).

% remove_induct
tff(fact_6304_remove__induct,axiom,
    ! [P2: fun_set_set_nat_bool,B4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(P2,bot_bot_set_set_nat))
     => ( ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B4))
         => pp(aa_set_set_nat_bool(P2,B4)) )
       => ( ! [A5: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A5))
             => ( ( A5 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A5),B4))
                 => ( ! [X3: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X3),A5))
                       => pp(aa_set_set_nat_bool(P2,minus_2163939370556025621et_nat(A5,insert_set_nat(X3,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P2,A5)) ) ) ) )
         => pp(aa_set_set_nat_bool(P2,B4)) ) ) ) ).

% remove_induct
tff(fact_6305_remove__induct,axiom,
    ! [P2: fun_set_complex_bool,B4: set_complex] :
      ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
     => ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
         => pp(aa_set_complex_bool(P2,B4)) )
       => ( ! [A5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
             => ( ( A5 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A5),B4))
                 => ( ! [X3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X3),A5))
                       => pp(aa_set_complex_bool(P2,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P2,A5)) ) ) ) )
         => pp(aa_set_complex_bool(P2,B4)) ) ) ) ).

% remove_induct
tff(fact_6306_remove__induct,axiom,
    ! [P2: fun_se8074476271188067385t_bool,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat))
     => ( ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
         => pp(aa_set3135850010578026034t_bool(P2,B4)) )
       => ( ! [A5: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
             => ( ( A5 != bot_bo2099793752762293965at_nat )
               => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A5),B4))
                 => ( ! [X3: product_prod_nat_nat] :
                        ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A5))
                       => pp(aa_set3135850010578026034t_bool(P2,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)))) )
                   => pp(aa_set3135850010578026034t_bool(P2,A5)) ) ) ) )
         => pp(aa_set3135850010578026034t_bool(P2,B4)) ) ) ) ).

% remove_induct
tff(fact_6307_remove__induct,axiom,
    ! [P2: fun_set_real_bool,B4: set_real] :
      ( pp(aa_set_real_bool(P2,bot_bot_set_real))
     => ( ( ~ pp(aa_set_real_bool(finite_finite_real,B4))
         => pp(aa_set_real_bool(P2,B4)) )
       => ( ! [A5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A5))
             => ( ( A5 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A5),B4))
                 => ( ! [X3: real] :
                        ( pp(aa_set_real_bool(member_real(X3),A5))
                       => pp(aa_set_real_bool(P2,minus_minus_set_real(A5,insert_real(X3,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P2,A5)) ) ) ) )
         => pp(aa_set_real_bool(P2,B4)) ) ) ) ).

% remove_induct
tff(fact_6308_remove__induct,axiom,
    ! [P2: fun_set_nat_bool,B4: set_nat] :
      ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
     => ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B4))
         => pp(aa_set_nat_bool(P2,B4)) )
       => ( ! [A5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A5))
             => ( ( A5 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A5),B4))
                 => ( ! [X3: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X3),A5))
                       => pp(aa_set_nat_bool(P2,minus_minus_set_nat(A5,insert_nat(X3,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P2,A5)) ) ) ) )
         => pp(aa_set_nat_bool(P2,B4)) ) ) ) ).

% remove_induct
tff(fact_6309_remove__induct,axiom,
    ! [P2: fun_set_int_bool,B4: set_int] :
      ( pp(aa_set_int_bool(P2,bot_bot_set_int))
     => ( ( ~ pp(aa_set_int_bool(finite_finite_int,B4))
         => pp(aa_set_int_bool(P2,B4)) )
       => ( ! [A5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A5))
             => ( ( A5 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A5),B4))
                 => ( ! [X3: int] :
                        ( pp(aa_set_int_bool(member_int(X3),A5))
                       => pp(aa_set_int_bool(P2,minus_minus_set_int(A5,insert_int(X3,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P2,A5)) ) ) ) )
         => pp(aa_set_int_bool(P2,B4)) ) ) ) ).

% remove_induct
tff(fact_6310_finite__remove__induct,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,P2: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,B4))
     => ( pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat))
       => ( ! [A5: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,A5))
             => ( ( A5 != bot_bo228742789529271731at_nat )
               => ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A5),B4))
                 => ( ! [X3: produc3843707927480180839at_nat] :
                        ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X3),A5))
                       => pp(aa_set4148026973699692364t_bool(P2,minus_3314409938677909166at_nat(A5,insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat)))) )
                   => pp(aa_set4148026973699692364t_bool(P2,A5)) ) ) ) )
         => pp(aa_set4148026973699692364t_bool(P2,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_6311_finite__remove__induct,axiom,
    ! [B4: set_set_nat,P2: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B4))
     => ( pp(aa_set_set_nat_bool(P2,bot_bot_set_set_nat))
       => ( ! [A5: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A5))
             => ( ( A5 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A5),B4))
                 => ( ! [X3: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X3),A5))
                       => pp(aa_set_set_nat_bool(P2,minus_2163939370556025621et_nat(A5,insert_set_nat(X3,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P2,A5)) ) ) ) )
         => pp(aa_set_set_nat_bool(P2,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_6312_finite__remove__induct,axiom,
    ! [B4: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [A5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
             => ( ( A5 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A5),B4))
                 => ( ! [X3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X3),A5))
                       => pp(aa_set_complex_bool(P2,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P2,A5)) ) ) ) )
         => pp(aa_set_complex_bool(P2,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_6313_finite__remove__induct,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
     => ( pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat))
       => ( ! [A5: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
             => ( ( A5 != bot_bo2099793752762293965at_nat )
               => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A5),B4))
                 => ( ! [X3: product_prod_nat_nat] :
                        ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A5))
                       => pp(aa_set3135850010578026034t_bool(P2,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat)))) )
                   => pp(aa_set3135850010578026034t_bool(P2,A5)) ) ) ) )
         => pp(aa_set3135850010578026034t_bool(P2,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_6314_finite__remove__induct,axiom,
    ! [B4: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [A5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A5))
             => ( ( A5 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A5),B4))
                 => ( ! [X3: real] :
                        ( pp(aa_set_real_bool(member_real(X3),A5))
                       => pp(aa_set_real_bool(P2,minus_minus_set_real(A5,insert_real(X3,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P2,A5)) ) ) ) )
         => pp(aa_set_real_bool(P2,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_6315_finite__remove__induct,axiom,
    ! [B4: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [A5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A5))
             => ( ( A5 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A5),B4))
                 => ( ! [X3: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X3),A5))
                       => pp(aa_set_nat_bool(P2,minus_minus_set_nat(A5,insert_nat(X3,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P2,A5)) ) ) ) )
         => pp(aa_set_nat_bool(P2,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_6316_finite__remove__induct,axiom,
    ! [B4: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [A5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A5))
             => ( ( A5 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A5),B4))
                 => ( ! [X3: int] :
                        ( pp(aa_set_int_bool(member_int(X3),A5))
                       => pp(aa_set_int_bool(P2,minus_minus_set_int(A5,insert_int(X3,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P2,A5)) ) ) ) )
         => pp(aa_set_int_bool(P2,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_6317_finite__induct__select,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,P2: fun_se8353504738346895699t_bool] :
      ( pp(aa_set4148026973699692364t_bool(finite4343798906461161616at_nat,S2))
     => ( pp(aa_set4148026973699692364t_bool(P2,bot_bo228742789529271731at_nat))
       => ( ! [T4: set_Pr4329608150637261639at_nat] :
              ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le2604355607129572851at_nat,T4),S2))
             => ( pp(aa_set4148026973699692364t_bool(P2,T4))
               => ? [X3: produc3843707927480180839at_nat] :
                    ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X3),minus_3314409938677909166at_nat(S2,T4)))
                    & pp(aa_set4148026973699692364t_bool(P2,insert9069300056098147895at_nat(X3,T4))) ) ) )
         => pp(aa_set4148026973699692364t_bool(P2,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6318_finite__induct__select,axiom,
    ! [S2: set_complex,P2: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [T4: set_complex] :
              ( pp(aa_set_complex_bool(ord_less_set_complex(T4),S2))
             => ( pp(aa_set_complex_bool(P2,T4))
               => ? [X3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(S2,T4)))
                    & pp(aa_set_complex_bool(P2,insert_complex(X3,T4))) ) ) )
         => pp(aa_set_complex_bool(P2,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6319_finite__induct__select,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,P2: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
     => ( pp(aa_set3135850010578026034t_bool(P2,bot_bo2099793752762293965at_nat))
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,T4),S2))
             => ( pp(aa_set3135850010578026034t_bool(P2,T4))
               => ? [X3: product_prod_nat_nat] :
                    ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),minus_1356011639430497352at_nat(S2,T4)))
                    & pp(aa_set3135850010578026034t_bool(P2,insert8211810215607154385at_nat(X3,T4))) ) ) )
         => pp(aa_set3135850010578026034t_bool(P2,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6320_finite__induct__select,axiom,
    ! [S2: set_real,P2: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [T4: set_real] :
              ( pp(aa_set_real_bool(ord_less_set_real(T4),S2))
             => ( pp(aa_set_real_bool(P2,T4))
               => ? [X3: real] :
                    ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(S2,T4)))
                    & pp(aa_set_real_bool(P2,insert_real(X3,T4))) ) ) )
         => pp(aa_set_real_bool(P2,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6321_finite__induct__select,axiom,
    ! [S2: set_int,P2: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [T4: set_int] :
              ( pp(aa_set_int_bool(ord_less_set_int(T4),S2))
             => ( pp(aa_set_int_bool(P2,T4))
               => ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(S2,T4)))
                    & pp(aa_set_int_bool(P2,insert_int(X3,T4))) ) ) )
         => pp(aa_set_int_bool(P2,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6322_finite__induct__select,axiom,
    ! [S2: set_nat,P2: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [T4: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,T4),S2))
             => ( pp(aa_set_nat_bool(P2,T4))
               => ? [X3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X3),minus_minus_set_nat(S2,T4)))
                    & pp(aa_set_nat_bool(P2,insert_nat(X3,T4))) ) ) )
         => pp(aa_set_nat_bool(P2,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6323_psubset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le2604355607129572851at_nat,A4),insert9069300056098147895at_nat(X,B4)))
    <=> ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B4))
         => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le2604355607129572851at_nat,A4),B4)) )
        & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),B4))
         => ( ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A4))
             => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le2604355607129572851at_nat,minus_3314409938677909166at_nat(A4,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B4)) )
            & ( ~ pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(X),A4))
             => pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(ord_le1268244103169919719at_nat,A4),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6324_psubset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A4),insert_set_nat(X,B4)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X),B4))
         => pp(aa_set_set_nat_bool(ord_less_set_set_nat(A4),B4)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),B4))
         => ( ( pp(aa_set_set_nat_bool(member_set_nat(X),A4))
             => pp(aa_set_set_nat_bool(ord_less_set_set_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat))),B4)) )
            & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),A4))
             => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6325_psubset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A4),insert8211810215607154385at_nat(X,B4)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B4))
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A4),B4)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B4))
         => ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A4))
             => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,minus_1356011639430497352at_nat(A4,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B4)) )
            & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A4))
             => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6326_psubset__insert__iff,axiom,
    ! [A4: set_real,X: real,B4: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A4),insert_real(X,B4)))
    <=> ( ( pp(aa_set_real_bool(member_real(X),B4))
         => pp(aa_set_real_bool(ord_less_set_real(A4),B4)) )
        & ( ~ pp(aa_set_real_bool(member_real(X),B4))
         => ( ( pp(aa_set_real_bool(member_real(X),A4))
             => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A4,insert_real(X,bot_bot_set_real))),B4)) )
            & ( ~ pp(aa_set_real_bool(member_real(X),A4))
             => pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6327_psubset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A4),insert_nat(X,B4)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X),B4))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A4),B4)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),B4))
         => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,minus_minus_set_nat(A4,insert_nat(X,bot_bot_set_nat))),B4)) )
            & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6328_psubset__insert__iff,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A4),insert_int(X,B4)))
    <=> ( ( pp(aa_set_int_bool(member_int(X),B4))
         => pp(aa_set_int_bool(ord_less_set_int(A4),B4)) )
        & ( ~ pp(aa_set_int_bool(member_int(X),B4))
         => ( ( pp(aa_set_int_bool(member_int(X),A4))
             => pp(aa_set_int_bool(ord_less_set_int(minus_minus_set_int(A4,insert_int(X,bot_bot_set_int))),B4)) )
            & ( ~ pp(aa_set_int_bool(member_int(X),A4))
             => pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6329_set__replicate__Suc,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] : set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(aa_nat_nat(suc,N),X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ).

% set_replicate_Suc
tff(fact_6330_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_6331_set__replicate__Suc,axiom,
    ! [N: nat,X: product_prod_nat_nat] : set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(aa_nat_nat(suc,N),X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ).

% set_replicate_Suc
tff(fact_6332_set__replicate__Suc,axiom,
    ! [N: nat,X: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X)) = insert_real(X,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_6333_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X)) = insert_nat(X,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_6334_set__replicate__Suc,axiom,
    ! [N: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X)) = insert_int(X,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_6335_set__replicate__conv__if,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = bot_bo228742789529271731at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6336_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_6337_set__replicate__conv__if,axiom,
    ! [N: nat,X: product_prod_nat_nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = bot_bo2099793752762293965at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6338_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_6339_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6340_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_6341_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( 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,M2),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M2)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_la(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6342_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M2)),groups2906978787729119204at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6343_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( 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,M2),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M2)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6344_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M2)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6345_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lc(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M2)) ) ) ).

% sum_Suc_diff
tff(fact_6346_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_ld(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,M2)) ) ) ).

% sum_Suc_diff
tff(fact_6347_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_le(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M2)) ) ) ).

% sum_Suc_diff
tff(fact_6348_ceiling__split,axiom,
    ! [P2: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P2,aa_real_int(archim7802044766580827645g_real,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I4))) )
         => pp(aa_int_bool(P2,I4)) ) ) ).

% ceiling_split
tff(fact_6349_ceiling__split,axiom,
    ! [P2: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P2,aa_rat_int(archim2889992004027027881ng_rat,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I4))) )
         => pp(aa_int_bool(P2,I4)) ) ) ).

% ceiling_split
tff(fact_6350_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))) ) ) ).

% ceiling_eq_iff
tff(fact_6351_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))) ) ) ).

% ceiling_eq_iff
tff(fact_6352_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_6353_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_6354_ceiling__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))),one_one_real)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ).

% ceiling_correct
tff(fact_6355_ceiling__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ) ).

% ceiling_correct
tff(fact_6356_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le
tff(fact_6357_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le
tff(fact_6358_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).

% ceiling_less_iff
tff(fact_6359_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% ceiling_less_iff
tff(fact_6360_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% le_ceiling_iff
tff(fact_6361_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).

% le_ceiling_iff
tff(fact_6362_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = 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,M2),N))),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,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_6363_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_6364_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = 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,M2),N))),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,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_6365_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),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,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_6366_sum_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6367_sum_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6368_sum_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6369_sum_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6370_sum_Oremove,axiom,
    ! [A4: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1935376822645274424al_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_6371_sum_Oremove,axiom,
    ! [A4: set_real,X: real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1932886352136224148al_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_6372_sum_Oremove,axiom,
    ! [A4: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups8097168146408367636l_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_6373_sum_Oremove,axiom,
    ! [A4: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1300246762558778688al_rat(G,A4) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_6374_sum_Oremove,axiom,
    ! [A4: set_int,X: int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( groups4541462559716669496nt_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_6375_sum_Oremove,axiom,
    ! [A4: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( groups8778361861064173332t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_6376_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_nat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5693394587270226106ex_nat(G,insert_complex(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6377_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_int,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5690904116761175830ex_int(G,insert_complex(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6378_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_real,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6379_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_rat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(G,insert_complex(X,A4)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6380_sum_Oinsert__remove,axiom,
    ! [A4: set_real,G: fun_real_nat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1935376822645274424al_nat(G,insert_real(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6381_sum_Oinsert__remove,axiom,
    ! [A4: set_real,G: fun_real_int,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1932886352136224148al_int(G,insert_real(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X)),groups1932886352136224148al_int(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6382_sum_Oinsert__remove,axiom,
    ! [A4: set_real,G: fun_real_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(G,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6383_sum_Oinsert__remove,axiom,
    ! [A4: set_real,G: fun_real_rat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(G,insert_real(X,A4)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X)),groups1300246762558778688al_rat(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6384_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_nat,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups4541462559716669496nt_nat(G,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(G,X)),groups4541462559716669496nt_nat(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_6385_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_real,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_6386_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A4)),aa_complex_int(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups5690904116761175830ex_int(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6387_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(A),A4))
         => ( groups1932886352136224148al_int(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(F,A4)),aa_real_int(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A4))
         => ( groups1932886352136224148al_int(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups1932886352136224148al_int(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6388_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A4)),aa_complex_real(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups5808333547571424918x_real(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6389_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(A),A4))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A4)),aa_real_real(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A4))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups8097168146408367636l_real(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6390_sum__diff1,axiom,
    ! [A4: set_int,A: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(A),A4))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A4)),aa_int_real(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A4))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups8778361861064173332t_real(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6391_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A4)),aa_complex_rat(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups5058264527183730370ex_rat(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6392_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(A),A4))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(F,A4)),aa_real_rat(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A4))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups1300246762558778688al_rat(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6393_sum__diff1,axiom,
    ! [A4: set_int,A: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(A),A4))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A4)),aa_int_rat(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A4))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups3906332499630173760nt_rat(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6394_sum__diff1,axiom,
    ! [A4: set_nat,A: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(A),A4))
         => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),aa_nat_int(F,A)) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),A4))
         => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_int(groups3539618377306564664at_int(F),A4) ) ) ) ) ).

% sum_diff1
tff(fact_6395_sum__diff1,axiom,
    ! [A4: set_nat,A: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(A),A4))
         => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A4)),aa_nat_rat(F,A)) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),A4))
         => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = groups2906978787729119204at_rat(F,A4) ) ) ) ) ).

% sum_diff1
tff(fact_6396_sum__count__set,axiom,
    ! [Xs: list_complex,X6: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),X6))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,X6))
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X6) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_6397_sum__count__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Xs)),X6))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X6))
       => ( groups977919841031483927at_nat(count_4203492906077236349at_nat(Xs),X6) = size_s5460976970255530739at_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_6398_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X6)
     => ( finite5795047828879050333T_VEBT(X6)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X6) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_6399_sum__count__set,axiom,
    ! [Xs: list_bool,X6: set_bool] :
      ( ord_less_eq_set_bool(set_bool2(Xs),X6)
     => ( finite_finite_bool(X6)
       => ( groups5768501273728744877ol_nat(count_list_bool(Xs),X6) = size_size_list_bool(Xs) ) ) ) ).

% sum_count_set
tff(fact_6400_sum__count__set,axiom,
    ! [Xs: list_int,X6: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),X6))
     => ( pp(aa_set_int_bool(finite_finite_int,X6))
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X6) = size_size_list_int(Xs) ) ) ) ).

% sum_count_set
tff(fact_6401_sum__count__set,axiom,
    ! [Xs: list_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),X6))
     => ( pp(aa_set_nat_bool(finite_finite_nat,X6))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X6) = size_size_list_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_6402_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_6403_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat,C2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_lf(A),B),C2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_lf(A),B),C2),S2) = groups5693394587270226106ex_nat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6404_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int,C2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lg(A),B),C2),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,A)),groups5690904116761175830ex_int(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lg(A),B),C2),S2) = groups5690904116761175830ex_int(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6405_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_lh(A),B),C2),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_lh(A),B),C2),S2) = groups5808333547571424918x_real(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6406_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_li(A),B),C2),S2) = aa_rat_rat(plus_plus_rat(aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_li(A),B),C2),S2) = groups5058264527183730370ex_rat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6407_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat,C2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_lj(A),B),C2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B,A)),groups1935376822645274424al_nat(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_lj(A),B),C2),S2) = groups1935376822645274424al_nat(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6408_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_int,C2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_lk(A),B),C2),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(B,A)),groups1932886352136224148al_int(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_lk(A),B),C2),S2) = groups1932886352136224148al_int(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6409_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ll(A),B),C2),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B,A)),groups8097168146408367636l_real(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ll(A),B),C2),S2) = groups8097168146408367636l_real(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6410_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lm(A),B),C2),S2) = aa_rat_rat(plus_plus_rat(aa_real_rat(B,A)),groups1300246762558778688al_rat(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lm(A),B),C2),S2) = groups1300246762558778688al_rat(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6411_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat,C2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ln(A),B),C2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(B,A)),groups4541462559716669496nt_nat(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ln(A),B),C2),S2) = groups4541462559716669496nt_nat(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6412_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_real,C2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lo(A),B),C2),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(B,A)),groups8778361861064173332t_real(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lo(A),B),C2),S2) = groups8778361861064173332t_real(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6413_ceiling__divide__upper,axiom,
    ! [Q2: real,P: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,P),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P),Q2)))),Q2))) ) ).

% ceiling_divide_upper
tff(fact_6414_ceiling__divide__upper,axiom,
    ! [Q2: rat,P: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P),Q2)))),Q2))) ) ).

% ceiling_divide_upper
tff(fact_6415_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_6416_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I2),A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A4,insert_complex(I2,bot_bot_set_complex))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),groups5808333547571424918x_real(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6417_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I2),A4))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A4,insert_real(I2,bot_bot_set_real))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),groups8097168146408367636l_real(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6418_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I2),A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A4,insert_int(I2,bot_bot_set_int))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),groups8778361861064173332t_real(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6419_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(member_complex(I2),A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A4,insert_complex(I2,bot_bot_set_complex))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),groups5058264527183730370ex_rat(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6420_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I2),A4))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A4,insert_real(I2,bot_bot_set_real))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),groups1300246762558778688al_rat(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6421_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I2),A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A4,insert_int(I2,bot_bot_set_int))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),groups3906332499630173760nt_rat(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6422_member__le__sum,axiom,
    ! [I2: nat,A4: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(member_nat(I2),A4))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(A4,insert_nat(I2,bot_bot_set_nat))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),groups2906978787729119204at_rat(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6423_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(member_complex(I2),A4))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A4,insert_complex(I2,bot_bot_set_complex))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),groups5693394587270226106ex_nat(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6424_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(member_real(I2),A4))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A4,insert_real(I2,bot_bot_set_real))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),groups1935376822645274424al_nat(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6425_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(member_int(I2),A4))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A4,insert_int(I2,bot_bot_set_int))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),groups4541462559716669496nt_nat(F,A4))) ) ) ) ).

% member_le_sum
tff(fact_6426_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_lp(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(F,M2)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_lp(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_6427_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lq(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M2)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lq(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_6428_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_lr(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M2)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_lr(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_6429_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ls(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M2)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ls(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_6430_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lt(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M2)) ) ) ).

% sum_telescope''
tff(fact_6431_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_lu(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,N)),aa_nat_rat(F,M2)) ) ) ).

% sum_telescope''
tff(fact_6432_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lv(F)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M2)) ) ) ).

% sum_telescope''
tff(fact_6433_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6434_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,N)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6435_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),nat_set_decode(Z)))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_6436_ceiling__divide__lower,axiom,
    ! [Q2: rat,P: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P),Q2)))),one_one_rat)),Q2)),P)) ) ).

% ceiling_divide_lower
tff(fact_6437_ceiling__divide__lower,axiom,
    ! [Q2: real,P: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P),Q2)))),one_one_real)),Q2)),P)) ) ).

% ceiling_divide_lower
tff(fact_6438_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6439_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,N)),one_one_rat)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6440_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = aa_set_nat_int(groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,N))) ).

% mask_eq_sum_exp
tff(fact_6441_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,N))) ).

% mask_eq_sum_exp
tff(fact_6442_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M2)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6443_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M2)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6444_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M2)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6445_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6446_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M2: nat,N: 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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lw(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.in_pairs
tff(fact_6447_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M2: nat,N: nat] : groups2906978787729119204at_rat(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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_lx(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.in_pairs
tff(fact_6448_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M2: nat,N: 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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ly(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.in_pairs
tff(fact_6449_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M2: nat,N: 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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lz(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% sum.in_pairs
tff(fact_6450_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,N))) ).

% mask_eq_sum_exp_nat
tff(fact_6451_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_6452_sum__gp,axiom,
    ! [N: nat,M2: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( ( ( X = one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = 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,N),one_one_nat)),M2)) ) )
          & ( ( X != one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M2)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_6453_sum__gp,axiom,
    ! [N: nat,M2: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( ( ( X = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M2)) ) )
          & ( ( X != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M2)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_6454_sum__gp,axiom,
    ! [N: nat,M2: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( ( ( X = one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = 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,N),one_one_nat)),M2)) ) )
          & ( ( X != one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_6455_gauss__sum__from__Suc__0,axiom,
    ! [N: 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)),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6456_gauss__sum__from__Suc__0,axiom,
    ! [N: 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)),N)) = 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,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6457_sum__gp__offset,axiom,
    ! [X: complex,M2: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M2)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6458_sum__gp__offset,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M2)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6459_sum__gp__offset,axiom,
    ! [X: real,M2: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6460_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6461_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6462_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6463_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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)),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6464_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6465_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(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)),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6466_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6467_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = 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,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6468_arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ma(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = 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,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6469_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mb(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = 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,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6470_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M2 = N ) ) ).

% of_nat_eq_iff
tff(fact_6471_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M2 = N ) ) ).

% of_nat_eq_iff
tff(fact_6472_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M2) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M2 = N ) ) ).

% of_nat_eq_iff
tff(fact_6473_int__eq__iff__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M2 = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_6474_negative__eq__positive,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M2) )
    <=> ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_6475_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% of_int_of_nat_eq
tff(fact_6476_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% of_int_of_nat_eq
tff(fact_6477_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% of_int_of_nat_eq
tff(fact_6478_negative__zle,axiom,
    ! [N: nat,M2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M2))) ).

% negative_zle
tff(fact_6479_int__dvd__int__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ).

% int_dvd_int_iff
tff(fact_6480_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M2) = zero_zero_complex )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6481_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M2) = zero_zero_rat )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6482_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = zero_zero_int )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6483_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = zero_zero_real )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6484_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M2) = zero_zero_nat )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6485_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_6486_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_6487_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_6488_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_6489_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_6490_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_6491_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_6492_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_6493_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_6494_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_6495_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_6496_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_6497_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_6498_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_6499_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_6500_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_6501_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_6502_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_6503_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_6504_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_6505_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_6506_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_6507_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M2)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_mult
tff(fact_6508_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_6509_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_6510_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_6511_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_6512_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_6513_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6514_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6515_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6516_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6517_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6518_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6519_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6520_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6521_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6522_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6523_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6524_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6525_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6526_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6527_negative__zless,axiom,
    ! [N: nat,M2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M2))) ).

% negative_zless
tff(fact_6528_of__nat__of__bool,axiom,
    ! [P2: bool] : aa_nat_real(semiri5074537144036343181t_real,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)) = zero_n3304061248610475627l_real(P2) ).

% of_nat_of_bool
tff(fact_6529_of__nat__of__bool,axiom,
    ! [P2: bool] : aa_nat_nat(semiri1316708129612266289at_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)) = aa_bool_nat(zero_n2687167440665602831ol_nat,P2) ).

% of_nat_of_bool
tff(fact_6530_of__nat__of__bool,axiom,
    ! [P2: bool] : aa_nat_int(semiri1314217659103216013at_int,aa_bool_nat(zero_n2687167440665602831ol_nat,P2)) = aa_bool_int(zero_n2684676970156552555ol_int,P2) ).

% of_nat_of_bool
tff(fact_6531_of__nat__sum,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A4)) = groups4538972089207619220nt_int(aTP_Lamm_mc(F),A4) ).

% of_nat_sum
tff(fact_6532_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A4: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A4)) = groups7754918857620584856omplex(aTP_Lamm_md(F),A4) ).

% of_nat_sum
tff(fact_6533_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_me(F)),A4) ).

% of_nat_sum
tff(fact_6534_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mf(F)),A4) ).

% of_nat_sum
tff(fact_6535_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mg(F)),A4) ).

% of_nat_sum
tff(fact_6536_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),zero_zero_real))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6537_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),zero_zero_rat))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6538_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),zero_zero_nat))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6539_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),zero_zero_int))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6540_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M2)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M2)) ).

% of_nat_Suc
tff(fact_6541_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M2)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M2)) ).

% of_nat_Suc
tff(fact_6542_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M2)) ).

% of_nat_Suc
tff(fact_6543_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M2)) ).

% of_nat_Suc
tff(fact_6544_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M2)) ).

% of_nat_Suc
tff(fact_6545_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_6546_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_num_real(numeral_numeral_real,W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W2))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_6547_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_nat_real(semiri5074537144036343181t_real,M2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M2)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_6548_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_6549_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_6550_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_6551_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_6552_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6553_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6554_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6555_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6556_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6557_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6558_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6559_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6560_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6561_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6562_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6563_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6564_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6565_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6566_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6567_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6568_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6569_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6570_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6571_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6572_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6573_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6574_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6575_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6576_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6577_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I2)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6578_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6579_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6580_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6581_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6582_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6583_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6584_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6585_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6586_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6587_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I2)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6588_real__arch__simple,axiom,
    ! [X: real] :
    ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N3))) ).

% real_arch_simple
tff(fact_6589_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N3))) ).

% real_arch_simple
tff(fact_6590_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N3))) ).

% reals_Archimedean2
tff(fact_6591_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N3))) ).

% reals_Archimedean2
tff(fact_6592_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).

% mult_of_nat_commute
tff(fact_6593_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_6594_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_6595_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_6596_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_6597_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N3: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% int_cases2
tff(fact_6598_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N3: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N3)) ).

% int_diff_cases
tff(fact_6599_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_6600_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_6601_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_6602_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_6603_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% of_nat_0_le_iff
tff(fact_6604_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_6605_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_6606_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_6607_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_6608_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_6609_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_6610_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_6611_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_6612_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_6613_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_6614_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_6615_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_6616_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_6617_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_6618_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_6619_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_6620_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_6621_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_6622_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_6623_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I2)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_6624_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I2)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_6625_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I2)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_6626_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_6627_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_6628_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(2)
tff(fact_6629_int__of__nat__induct,axiom,
    ! [P2: fun_int_bool,Z: int] :
      ( ! [N3: nat] : pp(aa_int_bool(P2,aa_nat_int(semiri1314217659103216013at_int,N3)))
     => ( ! [N3: nat] : pp(aa_int_bool(P2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3)))))
       => pp(aa_int_bool(P2,Z)) ) ) ).

% int_of_nat_induct
tff(fact_6630_int__cases,axiom,
    ! [Z: int] :
      ( ! [N3: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% int_cases
tff(fact_6631_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(3)
tff(fact_6632_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% zle_int
tff(fact_6633_zero__le__imp__eq__int,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
     => ? [N3: nat] : K2 = aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% zero_le_imp_eq_int
tff(fact_6634_nonneg__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
     => ~ ! [N3: nat] : K2 != aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% nonneg_int_cases
tff(fact_6635_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_6636_int__plus,axiom,
    ! [N: nat,M2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M2)) ).

% int_plus
tff(fact_6637_zadd__int__left,axiom,
    ! [M2: nat,N: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),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,M2),N))),Z) ).

% zadd_int_left
tff(fact_6638_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z))
    <=> ? [N2: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% zle_iff_zadd
tff(fact_6639_not__int__zless__negative,axiom,
    ! [N: nat,M2: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M2)))) ).

% not_int_zless_negative
tff(fact_6640_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_6641_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_6642_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_6643_nat__less__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_6644_nat__leq__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_6645_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N3)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_6646_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_6647_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_6648_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_6649_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_6650_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_6651_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y4: real] :
        ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),X))) ) ).

% reals_Archimedean3
tff(fact_6652_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N3: nat] : M2 != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( M2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ) ).

% int_cases4
tff(fact_6653_int__zle__neg,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M2))))
    <=> ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_6654_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
    <=> ? [N2: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% zless_iff_Suc_zadd
tff(fact_6655_negative__zle__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).

% negative_zle_0
tff(fact_6656_nonpos__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),zero_zero_int))
     => ~ ! [N3: nat] : K2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% nonpos_int_cases
tff(fact_6657_int__sum,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A4)) = groups4538972089207619220nt_int(aTP_Lamm_mc(F),A4) ).

% int_sum
tff(fact_6658_int__sum,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_me(F)),A4) ).

% int_sum
tff(fact_6659_mod__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,M2)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M2))) ).

% mod_mult2_eq'
tff(fact_6660_mod__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N: 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,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = 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,M2)),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,M2))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M2))) ).

% mod_mult2_eq'
tff(fact_6661_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
          & ( K2 = aa_nat_int(semiri1314217659103216013at_int,N3) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_6662_pos__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ~ ! [N3: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% pos_int_cases
tff(fact_6663_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) )
       => ~ ! [N3: nat] :
              ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ) ).

% int_cases3
tff(fact_6664_nat__less__real__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M2))) ) ).

% nat_less_real_le
tff(fact_6665_nat__le__real__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M2)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_6666_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_6667_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).

% not_zle_0_negative
tff(fact_6668_negative__zless__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_6669_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N3: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% negD
tff(fact_6670_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).

% int_ops(6)
tff(fact_6671_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M2),N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_6672_simp__from__to,axiom,
    ! [J: int,I2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I2),J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I2),J) = insert_int(I2,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J)) ) ) ) ).

% simp_from_to
tff(fact_6673_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
     => ~ ! [N3: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N3)))),E2)) ) ).

% nat_approx_posE
tff(fact_6674_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [N3: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))),E2)) ) ).

% nat_approx_posE
tff(fact_6675_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6676_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6677_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6678_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( ( N != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_6679_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( ( N != zero_zero_nat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M2))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_6680_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => ( ! [M: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),X)),C2)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_6681_neg__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
     => ~ ! [N3: nat] :
            ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% neg_int_cases
tff(fact_6682_zdiff__int__split,axiom,
    ! [P2: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P2,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P2,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_6683_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_6684_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_6685_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_6686_double__gauss__sum,axiom,
    ! [N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_6687_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_6688_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum
tff(fact_6689_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum
tff(fact_6690_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_mh(A),D),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D))) ).

% double_arith_series
tff(fact_6691_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_mi(A),D),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_6692_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_mj(A),D),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),D))) ).

% double_arith_series
tff(fact_6693_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ma(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = 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,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_6694_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mb(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = 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,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).

% double_arith_series
tff(fact_6695_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mk(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D))) ).

% double_arith_series
tff(fact_6696_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(semiri8010041392384452111omplex,N) = zero_zero_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(semiri8010041392384452111omplex,N) = produc1917071388513777916omplex(aTP_Lamm_ml,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6697_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(semiri681578069525770553at_rat,N) = zero_zero_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(semiri681578069525770553at_rat,N) = produc6207742614233964070at_rat(aTP_Lamm_mm,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6698_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,N) = produc2676513652042109336d_enat(aTP_Lamm_mn,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6699_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(semiri1314217659103216013at_int,N) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_mo),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6700_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_mp),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6701_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(semiri1316708129612266289at_nat,N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_mq),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6702_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_6703_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_6704_ln__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_mr,X)) ) ) ) ).

% ln_series
tff(fact_6705_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,N: 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)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mt(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6706_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),N)),aa_nat_rat(power_power_rat(Z),N))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mv(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6707_lemma__termdiff2,axiom,
    ! [H: real,Z: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mx(H),Z),N)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6708_pochhammer__double,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_6709_pochhammer__double,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_6710_pochhammer__double,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_6711_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_my),N),zero_zero_complex) ).

% of_nat_code
tff(fact_6712_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_mz,N),zero_zero_rat) ).

% of_nat_code
tff(fact_6713_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_na),N),zero_zero_int) ).

% of_nat_code
tff(fact_6714_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_nb),N),zero_zero_real) ).

% of_nat_code
tff(fact_6715_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_nc),N),zero_zero_nat) ).

% of_nat_code
tff(fact_6716_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,X) = aa_nat_set_nat(set_ord_lessThan_nat,Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_6717_lessThan__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( set_ord_lessThan_int(X) = set_ord_lessThan_int(Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_6718_lessThan__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( set_or5984915006950818249n_real(X) = set_or5984915006950818249n_real(Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_6719_lessThan__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I2),set_or890127255671739683et_nat(K2)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,I2),K2)) ) ).

% lessThan_iff
tff(fact_6720_lessThan__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( pp(aa_set_rat_bool(member_rat(I2),set_ord_lessThan_rat(K2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),K2)) ) ).

% lessThan_iff
tff(fact_6721_lessThan__iff,axiom,
    ! [I2: num,K2: num] :
      ( pp(aa_set_num_bool(member_num(I2),set_ord_lessThan_num(K2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I2),K2)) ) ).

% lessThan_iff
tff(fact_6722_lessThan__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_set_nat_bool(member_nat(I2),aa_nat_set_nat(set_ord_lessThan_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K2)) ) ).

% lessThan_iff
tff(fact_6723_lessThan__iff,axiom,
    ! [I2: int,K2: int] :
      ( pp(aa_set_int_bool(member_int(I2),set_ord_lessThan_int(K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K2)) ) ).

% lessThan_iff
tff(fact_6724_lessThan__iff,axiom,
    ! [I2: real,K2: real] :
      ( pp(aa_set_real_bool(member_real(I2),set_or5984915006950818249n_real(K2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),K2)) ) ).

% lessThan_iff
tff(fact_6725_finite__lessThan,axiom,
    ! [K2: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K2))) ).

% finite_lessThan
tff(fact_6726_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_lessThan_rat(X)),set_ord_lessThan_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6727_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_lessThan_num(X)),set_ord_lessThan_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6728_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6729_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6730_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X)),set_or5984915006950818249n_real(Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6731_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_6732_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_6733_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_6734_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_6735_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_6736_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_6737_single__Diff__lessThan,axiom,
    ! [K2: nat] : minus_minus_set_nat(insert_nat(K2,bot_bot_set_nat),aa_nat_set_nat(set_ord_lessThan_nat,K2)) = insert_nat(K2,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_6738_single__Diff__lessThan,axiom,
    ! [K2: int] : minus_minus_set_int(insert_int(K2,bot_bot_set_int),set_ord_lessThan_int(K2)) = insert_int(K2,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_6739_single__Diff__lessThan,axiom,
    ! [K2: real] : minus_minus_set_real(insert_real(K2,bot_bot_set_real),set_or5984915006950818249n_real(K2)) = insert_real(K2,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_6740_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_int(G,N)) ).

% sum.lessThan_Suc
tff(fact_6741_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ).

% sum.lessThan_Suc
tff(fact_6742_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_nat(G,N)) ).

% sum.lessThan_Suc
tff(fact_6743_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% sum.lessThan_Suc
tff(fact_6744_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_jr(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_6745_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_jt(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_6746_int__int__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M2 = N ) ) ).

% int_int_eq
tff(fact_6747_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_6748_lessThan__non__empty,axiom,
    ! [X: real] : set_or5984915006950818249n_real(X) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_6749_infinite__Iio,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A))) ).

% infinite_Iio
tff(fact_6750_infinite__Iio,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(finite_finite_real,set_or5984915006950818249n_real(A))) ).

% infinite_Iio
tff(fact_6751_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_nd,U)) ).

% lessThan_def
tff(fact_6752_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_ne,U)) ).

% lessThan_def
tff(fact_6753_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_nf,U)) ).

% lessThan_def
tff(fact_6754_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,U)) ).

% lessThan_def
tff(fact_6755_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_ng,U)) ).

% lessThan_def
tff(fact_6756_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_nh,U)) ).

% lessThan_def
tff(fact_6757_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_6758_lessThan__strict__subset__iff,axiom,
    ! [M2: rat,N: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_ord_lessThan_rat(M2)),set_ord_lessThan_rat(N)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6759_lessThan__strict__subset__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_ord_lessThan_num(M2)),set_ord_lessThan_num(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6760_lessThan__strict__subset__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M2)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6761_lessThan__strict__subset__iff,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M2)),set_ord_lessThan_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6762_lessThan__strict__subset__iff,axiom,
    ! [M2: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M2)),set_or5984915006950818249n_real(N)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6763_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).

% pochhammer_pos
tff(fact_6764_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_pos
tff(fact_6765_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_pos
tff(fact_6766_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_pos
tff(fact_6767_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M2) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_6768_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M2) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_6769_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M2) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_6770_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M2) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_6771_pochhammer__neq__0__mono,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M2) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_6772_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M2) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_6773_lessThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = insert_nat(K2,aa_nat_set_nat(set_ord_lessThan_nat,K2)) ).

% lessThan_Suc
tff(fact_6774_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_6775_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [K: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ).

% finite_nat_bounded
tff(fact_6776_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K3))) ) ).

% finite_nat_iff_bounded
tff(fact_6777_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).

% pochhammer_nonneg
tff(fact_6778_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_nonneg
tff(fact_6779_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_nonneg
tff(fact_6780_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_nonneg
tff(fact_6781_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_6782_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_6783_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% pochhammer_0_left
tff(fact_6784_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_6785_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_6786_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K2))) ).

% lessThan_nat_numeral
tff(fact_6787_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ni(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_6788_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nj(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_6789_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P2: fun_int_nat,N: int] :
      ( ! [X4: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(Q,X4)),aa_int_nat(P2,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P2,set_ord_lessThan_int(N))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(N))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ko(Q),P2),set_ord_lessThan_int(N)) ) ) ).

% sum_diff_distrib
tff(fact_6790_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P2: fun_real_nat,N: real] :
      ( ! [X4: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(Q,X4)),aa_real_nat(P2,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P2,set_or5984915006950818249n_real(N))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(N))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_km(Q),P2),set_or5984915006950818249n_real(N)) ) ) ).

% sum_diff_distrib
tff(fact_6791_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P2: fun_nat_nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P2,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kp(Q),P2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_6792_pochhammer__rec,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),N)) ).

% pochhammer_rec
tff(fact_6793_pochhammer__rec,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_6794_pochhammer__rec,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),N)) ).

% pochhammer_rec
tff(fact_6795_pochhammer__rec,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = 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)),N)) ).

% pochhammer_rec
tff(fact_6796_pochhammer__rec,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = 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)),N)) ).

% pochhammer_rec
tff(fact_6797_pochhammer__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),N)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))) ).

% pochhammer_Suc
tff(fact_6798_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_6799_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_6800_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_6801_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_6802_pochhammer__rec_H,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N)) ).

% pochhammer_rec'
tff(fact_6803_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).

% pochhammer_rec'
tff(fact_6804_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = 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,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).

% pochhammer_rec'
tff(fact_6805_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)) ).

% pochhammer_rec'
tff(fact_6806_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = 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,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).

% pochhammer_rec'
tff(fact_6807_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K2) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6808_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K2) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6809_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K2) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6810_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K2) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6811_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K2) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6812_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K2) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6813_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K2) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6814_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K2) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6815_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6816_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6817_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6818_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K2) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6819_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K2) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6820_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K2) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6821_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K2) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6822_pochhammer__product_H,axiom,
    ! [Z: complex,N: nat,M2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),N)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),M2)) ).

% pochhammer_product'
tff(fact_6823_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M2)) ).

% pochhammer_product'
tff(fact_6824_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M2: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M2)) ).

% pochhammer_product'
tff(fact_6825_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M2)) ).

% pochhammer_product'
tff(fact_6826_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),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,N))),M2)) ).

% pochhammer_product'
tff(fact_6827_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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_la(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6828_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6829_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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_kq(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6830_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6831_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lc(F)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M2)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6832_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M2: nat] : groups2906978787729119204at_rat(aTP_Lamm_ld(F),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M2)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6833_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_le(F)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M2)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6834_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_nk(F)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M2)) ).

% sum_lessThan_telescope'
tff(fact_6835_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M2: nat] : groups2906978787729119204at_rat(aTP_Lamm_nl(F),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M2)) ).

% sum_lessThan_telescope'
tff(fact_6836_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nm(F)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M2)) ).

% sum_lessThan_telescope'
tff(fact_6837_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: 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)),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6838_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,N: 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)),N)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6839_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6840_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6841_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6842_one__diff__power__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6843_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6844_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6845_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6846_power__diff__1__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6847_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_6848_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N)),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_6849_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_6850_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),N) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),M2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_6851_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_6852_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_6853_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),M2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_6854_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M2)),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,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_6855_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6856_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6857_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6858_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M2: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(Z),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_no(Z),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6859_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M2: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_np(Z),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nq(Z),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6860_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nr(Z),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ns(Z),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6861_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nt(Z),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nu(Z),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6862_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nv(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6863_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nw(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6864_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nx(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6865_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ny(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6866_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nz(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6867_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_oa(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6868_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ob(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6869_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_oc(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6870_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K2)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K2)) ).

% pochhammer_absorb_comp
tff(fact_6871_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K2: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K2)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K2)) ).

% pochhammer_absorb_comp
tff(fact_6872_pochhammer__absorb__comp,axiom,
    ! [R2: int,K2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K2))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K2)) ).

% pochhammer_absorb_comp
tff(fact_6873_pochhammer__absorb__comp,axiom,
    ! [R2: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K2)) ).

% pochhammer_absorb_comp
tff(fact_6874_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K5: rat,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,P4)),K5)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K5))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6875_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K5: int,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,P4)),K5)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K5))
       => pp(aa_int_bool(aa_int_fun_int_bool(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,N),K2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6876_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K5: nat,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,P4)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6877_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K5: real,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,P4)),K5)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K5))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,N),K2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6878_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_od(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6879_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oe(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6880_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_of(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6881_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_og(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6882_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2)) ).

% pochhammer_minus'
tff(fact_6883_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2)) ).

% pochhammer_minus'
tff(fact_6884_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2)) ).

% pochhammer_minus'
tff(fact_6885_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2)) ).

% pochhammer_minus'
tff(fact_6886_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% pochhammer_minus
tff(fact_6887_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% pochhammer_minus
tff(fact_6888_pochhammer__minus,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2)) ).

% pochhammer_minus
tff(fact_6889_pochhammer__minus,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% pochhammer_minus
tff(fact_6890_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_oh(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oi(F)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oj(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_6891_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_6892_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_6893_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_6894_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_6895_suminf__geometric,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C2)),one_one_real))
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C2)) ) ) ).

% suminf_geometric
tff(fact_6896_suminf__geometric,axiom,
    ! [C2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C2)),one_one_real))
     => ( suminf_complex(power_power_complex(C2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C2)) ) ) ).

% suminf_geometric
tff(fact_6897_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_6898_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_6899_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_6900_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_6901_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_ok) = zero_zero_complex ).

% suminf_zero
tff(fact_6902_suminf__zero,axiom,
    suminf_real(aTP_Lamm_fr) = zero_zero_real ).

% suminf_zero
tff(fact_6903_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_bz) = zero_zero_nat ).

% suminf_zero
tff(fact_6904_suminf__zero,axiom,
    suminf_int(aTP_Lamm_ol) = zero_zero_int ).

% suminf_zero
tff(fact_6905_norm__not__less__zero,axiom,
    ! [X: complex] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)) ).

% norm_not_less_zero
tff(fact_6906_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% norm_uminus_minus
tff(fact_6907_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_6908_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6909_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6910_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6911_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = aa_nat_complex(power_power_complex(Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6912_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_6913_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_6914_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_6915_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_6916_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_6917_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_6918_norm__add__leD,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C2))) ) ).

% norm_add_leD
tff(fact_6919_norm__add__leD,axiom,
    ! [A: complex,B: complex,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C2))) ) ).

% norm_add_leD
tff(fact_6920_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_6921_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_6922_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_6923_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_6924_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_6925_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_6926_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_6927_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_6928_norm__diff__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))) ).

% norm_diff_ineq
tff(fact_6929_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B)))) ).

% norm_diff_ineq
tff(fact_6930_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_complex(F,N3) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N4) ) ) ) ).

% suminf_finite
tff(fact_6931_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_int(F,N3) = zero_zero_int ) )
       => ( suminf_int(F) = aa_set_nat_int(groups3539618377306564664at_int(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6932_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_nat(F,N3) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6933_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_real(F,N3) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N4) ) ) ) ).

% suminf_finite
tff(fact_6934_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6935_power__eq__1__iff,axiom,
    ! [W2: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6936_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C2: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_6937_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C2: complex,D: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(C2),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_6938_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups6464643781859351333omplex(aTP_Lamm_om(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,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6939_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups73079841787564623at_rat(aTP_Lamm_on(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,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6940_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_oo,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,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6941_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_op) ).

% pi_series
tff(fact_6942_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      ( ( ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
          & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K2),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
      & ( ~ ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_6943_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_6944_pochhammer__code,axiom,
    ! [N: nat,A: complex] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_oq(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_6945_pochhammer__code,axiom,
    ! [N: nat,A: rat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_or(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat) ) ) ) ).

% pochhammer_code
tff(fact_6946_pochhammer__code,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_os(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_6947_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ot(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_6948_pochhammer__code,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ou(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_6949_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_6950_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_6951_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_6952_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_6953_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_6954_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_6955_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_6956_of__nat__prod,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A4)) = groups1705073143266064639nt_int(aTP_Lamm_mc(F),A4) ).

% of_nat_prod
tff(fact_6957_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_mg(F)),A4) ).

% of_nat_prod
tff(fact_6958_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_me(F)),A4) ).

% of_nat_prod
tff(fact_6959_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mf(F)),A4) ).

% of_nat_prod
tff(fact_6960_of__int__prod,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_real(ring_1_of_int_real,aa_set_nat_int(groups705719431365010083at_int(F),A4)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gn(F)),A4) ).

% of_int_prod
tff(fact_6961_of__int__prod,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_rat(ring_1_of_int_rat,aa_set_nat_int(groups705719431365010083at_int(F),A4)) = groups73079841787564623at_rat(aTP_Lamm_ov(F),A4) ).

% of_int_prod
tff(fact_6962_of__int__prod,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_int_int(ring_1_of_int_int,aa_set_nat_int(groups705719431365010083at_int(F),A4)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ow(F)),A4) ).

% of_int_prod
tff(fact_6963_of__int__prod,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A4)) = groups2316167850115554303t_real(aTP_Lamm_go(F),A4) ).

% of_int_prod
tff(fact_6964_of__int__prod,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A4)) = groups1072433553688619179nt_rat(aTP_Lamm_gp(F),A4) ).

% of_int_prod
tff(fact_6965_of__int__prod,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A4)) = groups1705073143266064639nt_int(aTP_Lamm_gq(F),A4) ).

% of_int_prod
tff(fact_6966_prod__zero__iff,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( groups6464643781859351333omplex(F,A4) = zero_zero_complex )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_complex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6967_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups7440179247065528705omplex(F,A4) = zero_zero_complex )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_complex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6968_prod__zero__iff,axiom,
    ! [A4: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups3708469109370488835omplex(F,A4) = zero_zero_complex )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_complex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6969_prod__zero__iff,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A4) = zero_zero_real )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6970_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups2316167850115554303t_real(F,A4) = zero_zero_real )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6971_prod__zero__iff,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups766887009212190081x_real(F,A4) = zero_zero_real )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6972_prod__zero__iff,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( groups73079841787564623at_rat(F,A4) = zero_zero_rat )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_rat(F,X5) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6973_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups1072433553688619179nt_rat(F,A4) = zero_zero_rat )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_rat(F,X5) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6974_prod__zero__iff,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups225925009352817453ex_rat(F,A4) = zero_zero_rat )
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_rat(F,X5) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6975_prod__zero__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups1707563613775114915nt_nat(F,A4) = zero_zero_nat )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6976_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_6977_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_6978_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_6979_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_6980_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_6981_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_6982_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_6983_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_6984_prod_Oempty,axiom,
    ! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).

% prod.empty
tff(fact_6985_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_6986_prod_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1707563613775114915nt_nat(G,A4) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6987_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups861055069439313189ex_nat(G,A4) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6988_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_int] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups858564598930262913ex_int(G,A4) = one_one_int ) ) ).

% prod.infinite
tff(fact_6989_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_6990_prod_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups2316167850115554303t_real(G,A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_6991_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups766887009212190081x_real(G,A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_6992_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups6464643781859351333omplex(G,A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_6993_prod_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups7440179247065528705omplex(G,A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_6994_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups3708469109370488835omplex(G,A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_6995_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups73079841787564623at_rat(G,A4) = one_one_rat ) ) ).

% prod.infinite
tff(fact_6996_dvd__prod__eqI,axiom,
    ! [A4: set_real,A: real,B: nat,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => ( ( B = aa_real_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups4696554848551431203al_nat(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6997_dvd__prod__eqI,axiom,
    ! [A4: set_int,A: int,B: nat,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => ( ( B = aa_int_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups1707563613775114915nt_nat(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6998_dvd__prod__eqI,axiom,
    ! [A4: set_complex,A: complex,B: nat,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(A),A4))
       => ( ( B = aa_complex_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups861055069439313189ex_nat(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_6999_dvd__prod__eqI,axiom,
    ! [A4: set_real,A: real,B: int,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => ( ( B = aa_real_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups4694064378042380927al_int(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7000_dvd__prod__eqI,axiom,
    ! [A4: set_complex,A: complex,B: int,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(A),A4))
       => ( ( B = aa_complex_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups858564598930262913ex_int(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7001_dvd__prod__eqI,axiom,
    ! [A4: set_nat,A: nat,B: int,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => ( ( B = aa_nat_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),aa_set_nat_int(groups705719431365010083at_int(F),A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7002_dvd__prod__eqI,axiom,
    ! [A4: set_int,A: int,B: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => ( ( B = aa_int_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups1705073143266064639nt_int(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7003_dvd__prod__eqI,axiom,
    ! [A4: set_nat,A: nat,B: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => ( ( B = aa_nat_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),aa_set_nat_nat(groups708209901874060359at_nat(F),A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7004_dvd__prod__eqI,axiom,
    ! [A4: set_set_nat,A: set_nat,B: nat,F: fun_set_nat_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => ( ( B = aa_set_nat_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups4248547760180025341at_nat(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7005_dvd__prod__eqI,axiom,
    ! [A4: set_set_nat,A: set_nat,B: int,F: fun_set_nat_int] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => ( ( B = aa_set_nat_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups4246057289670975065at_int(F,A4))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7006_dvd__prodI,axiom,
    ! [A4: set_real,A: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7007_dvd__prodI,axiom,
    ! [A4: set_int,A: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7008_dvd__prodI,axiom,
    ! [A4: set_complex,A: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(A),A4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7009_dvd__prodI,axiom,
    ! [A4: set_real,A: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A)),groups4694064378042380927al_int(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7010_dvd__prodI,axiom,
    ! [A4: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(A),A4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A)),groups858564598930262913ex_int(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7011_dvd__prodI,axiom,
    ! [A4: set_nat,A: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(F,A)),aa_set_nat_int(groups705719431365010083at_int(F),A4))) ) ) ).

% dvd_prodI
tff(fact_7012_dvd__prodI,axiom,
    ! [A4: set_int,A: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(F,A)),groups1705073143266064639nt_int(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7013_dvd__prodI,axiom,
    ! [A4: set_nat,A: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(F,A)),aa_set_nat_nat(groups708209901874060359at_nat(F),A4))) ) ) ).

% dvd_prodI
tff(fact_7014_dvd__prodI,axiom,
    ! [A4: set_set_nat,A: set_nat,F: fun_set_nat_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(F,A)),groups4248547760180025341at_nat(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7015_dvd__prodI,axiom,
    ! [A4: set_set_nat,A: set_nat,F: fun_set_nat_int] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A4))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_set_nat_int(F,A)),groups4246057289670975065at_int(F,A4))) ) ) ).

% dvd_prodI
tff(fact_7016_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K2),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% and_negative_int_iff
tff(fact_7017_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ox(A),B),S2) = aa_real_nat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ox(A),B),S2) = one_one_nat ) ) ) ) ).

% prod.delta'
tff(fact_7018_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_oy(A),B),S2) = aa_int_nat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_oy(A),B),S2) = one_one_nat ) ) ) ) ).

% prod.delta'
tff(fact_7019_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oz(A),B),S2) = aa_complex_nat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oz(A),B),S2) = one_one_nat ) ) ) ) ).

% prod.delta'
tff(fact_7020_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_pa(A),B),S2) = aa_real_int(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_pa(A),B),S2) = one_one_int ) ) ) ) ).

% prod.delta'
tff(fact_7021_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_pb(A),B),S2) = aa_complex_int(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_pb(A),B),S2) = one_one_int ) ) ) ) ).

% prod.delta'
tff(fact_7022_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pc(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pc(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7023_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pd(A),B)),S2) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pd(A),B)),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7024_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_pe(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_pe(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7025_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pf(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pf(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7026_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_pg(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_pg(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7027_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ph(A),B),S2) = aa_real_nat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ph(A),B),S2) = one_one_nat ) ) ) ) ).

% prod.delta
tff(fact_7028_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pi(A),B),S2) = aa_int_nat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pi(A),B),S2) = one_one_nat ) ) ) ) ).

% prod.delta
tff(fact_7029_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_pj(A),B),S2) = aa_complex_nat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_pj(A),B),S2) = one_one_nat ) ) ) ) ).

% prod.delta
tff(fact_7030_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_pk(A),B),S2) = aa_real_int(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4694064378042380927al_int(aa_fun8270430325945277160al_int(aTP_Lamm_pk(A),B),S2) = one_one_int ) ) ) ) ).

% prod.delta
tff(fact_7031_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_pl(A),B),S2) = aa_complex_int(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups858564598930262913ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_pl(A),B),S2) = one_one_int ) ) ) ) ).

% prod.delta
tff(fact_7032_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pm(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pm(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7033_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pn(A),B)),S2) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pn(A),B)),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7034_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_po(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_po(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7035_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pp(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pp(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7036_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_pq(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_pq(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7037_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups713298508707869441omplex(G,insert_real(X,A4)) = aa_complex_complex(times_times_complex(aa_real_complex(G,X)),groups713298508707869441omplex(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7038_prod_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
       => ( groups6464643781859351333omplex(G,insert_nat(X,A4)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,X)),groups6464643781859351333omplex(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7039_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( groups7440179247065528705omplex(G,insert_int(X,A4)) = aa_complex_complex(times_times_complex(aa_int_complex(G,X)),groups7440179247065528705omplex(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7040_prod_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups3708469109370488835omplex(G,insert_complex(X,A4)) = aa_complex_complex(times_times_complex(aa_complex_complex(G,X)),groups3708469109370488835omplex(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7041_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1681761925125756287l_real(G,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7042_prod_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),aa_set_nat_real(groups129246275422532515t_real(G),A4)) ) ) ) ).

% prod.insert
tff(fact_7043_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( groups2316167850115554303t_real(G,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7044_prod_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups766887009212190081x_real(G,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7045_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups4061424788464935467al_rat(G,insert_real(X,A4)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X)),groups4061424788464935467al_rat(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7046_prod_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
       => ( groups73079841787564623at_rat(G,insert_nat(X,A4)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A4)) ) ) ) ).

% prod.insert
tff(fact_7047_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_complex(G,N)) ).

% prod.lessThan_Suc
tff(fact_7048_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% prod.lessThan_Suc
tff(fact_7049_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ).

% prod.lessThan_Suc
tff(fact_7050_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_int(G,N)) ).

% prod.lessThan_Suc
tff(fact_7051_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_nat(G,N)) ).

% prod.lessThan_Suc
tff(fact_7052_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_7053_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_7054_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_7055_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_7056_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_7057_log__eq__one,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_7058_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_7059_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_7060_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_7061_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_7062_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_7063_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_7064_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_7065_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_7066_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_7067_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7068_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7069_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7070_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = 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,M2),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7071_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = 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,M2),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7072_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_7073_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_7074_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_7075_prod_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_real_fun_nat_int,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups4694064378042380927al_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_pr(B4),G),R),A4) = aa_set_nat_int(groups705719431365010083at_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_pt(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7076_prod_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: fun_co7370231399432144932at_int,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups858564598930262913ex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_pu(B4),G),R),A4) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_pw(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7077_prod_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_int,G: fun_real_fun_int_int,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( groups4694064378042380927al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_px(B4),G),R),A4) = groups1705073143266064639nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_py(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7078_prod_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_int,G: fun_co333780937270390912nt_int,R: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B4))
       => ( groups858564598930262913ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_pz(B4),G),R),A4) = groups1705073143266064639nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_qa(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7079_prod_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups4696554848551431203al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_qb(B4),G),R),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_qc(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7080_prod_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups1707563613775114915nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_qd(B4),G),R),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_qe(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7081_prod_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( groups861055069439313189ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_qf(B4),G),R),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_qg(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7082_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_real,G: fun_nat_fun_real_int,R: fun_na6801867027684862029l_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_real_bool(finite_finite_real,B4))
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun3886942400138156042at_int(aa_fun699103053049729529at_int(aTP_Lamm_qi(B4),G),R)),A4) = groups4694064378042380927al_int(aa_fun5434943856969160678al_int(aa_fun2246985729214054869al_int(aTP_Lamm_ql(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7083_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_complex,G: fun_na5967208032043182372ex_int,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun3278734219653515656at_int(aa_fun9069282223758319861at_int(aTP_Lamm_qm(B4),G),R)),A4) = groups858564598930262913ex_int(aa_fun4682754999372510822ex_int(aa_fun2078238821050448595ex_int(aTP_Lamm_qo(A4),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7084_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_nat,G: fun_nat_fun_nat_int,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun467087377561674726at_int(aa_fun989774750125491633at_int(aTP_Lamm_qq(B4),G),R)),A4) = aa_set_nat_int(groups705719431365010083at_int(aa_fun467087377561674726at_int(aa_fun989774750125491633at_int(aTP_Lamm_qs(A4),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_7085_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I3)),aa_real_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(G,A4))) ) ).

% prod_mono
tff(fact_7086_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),aa_nat_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),aa_set_nat_real(groups129246275422532515t_real(G),A4))) ) ).

% prod_mono
tff(fact_7087_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I3)),aa_int_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(G,A4))) ) ).

% prod_mono
tff(fact_7088_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(G,A4))) ) ).

% prod_mono
tff(fact_7089_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(G,A4))) ) ).

% prod_mono
tff(fact_7090_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(G,A4))) ) ).

% prod_mono
tff(fact_7091_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A4)),groups4696554848551431203al_nat(G,A4))) ) ).

% prod_mono
tff(fact_7092_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A4)),groups1707563613775114915nt_nat(G,A4))) ) ).

% prod_mono
tff(fact_7093_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,I3)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3))) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(G,A4))) ) ).

% prod_mono
tff(fact_7094_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I3)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3))) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A4)),aa_set_nat_int(groups705719431365010083at_int(G),A4))) ) ).

% prod_mono
tff(fact_7095_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A4))) ) ).

% prod_nonneg
tff(fact_7096_prod__nonneg,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A4))) ) ).

% prod_nonneg
tff(fact_7097_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A4))) ) ).

% prod_nonneg
tff(fact_7098_prod__pos,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(F),A4))) ) ).

% prod_pos
tff(fact_7099_prod__pos,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A4))) ) ).

% prod_pos
tff(fact_7100_prod__pos,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A4))) ) ).

% prod_pos
tff(fact_7101_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A4))) ) ).

% prod_ge_1
tff(fact_7102_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),A4))) ) ).

% prod_ge_1
tff(fact_7103_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A4))) ) ).

% prod_ge_1
tff(fact_7104_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A4))) ) ).

% prod_ge_1
tff(fact_7105_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A4))) ) ).

% prod_ge_1
tff(fact_7106_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A4))) ) ).

% prod_ge_1
tff(fact_7107_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A4))) ) ).

% prod_ge_1
tff(fact_7108_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A4))) ) ).

% prod_ge_1
tff(fact_7109_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A4))) ) ).

% prod_ge_1
tff(fact_7110_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),A4))) ) ).

% prod_ge_1
tff(fact_7111_prod__zero,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
            & ( aa_nat_complex(F,X3) = zero_zero_complex ) )
       => ( groups6464643781859351333omplex(F,A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7112_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & ( aa_int_complex(F,X3) = zero_zero_complex ) )
       => ( groups7440179247065528705omplex(F,A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7113_prod__zero,axiom,
    ! [A4: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
            & ( aa_complex_complex(F,X3) = zero_zero_complex ) )
       => ( groups3708469109370488835omplex(F,A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7114_prod__zero,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
            & ( aa_nat_real(F,X3) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7115_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & ( aa_int_real(F,X3) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7116_prod__zero,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
            & ( aa_complex_real(F,X3) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7117_prod__zero,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
            & ( aa_nat_rat(F,X3) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7118_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & ( aa_int_rat(F,X3) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7119_prod__zero,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
            & ( aa_complex_rat(F,X3) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7120_prod__zero,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & ( aa_int_nat(F,X3) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A4) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_7121_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_qt(F),A),B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_7122_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups129246275422532515t_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_qu(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_7123_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_qv(F),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_7124_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups705719431365010083at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_qw(F),A),B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_7125_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_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_qx(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_7126_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_7127_pi__not__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),zero_zero_real)) ).

% pi_not_less_zero
tff(fact_7128_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_nat,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4696554848551431203al_nat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_il(A4),P2))) = groups4696554848551431203al_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_qy(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7129_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_nat,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1707563613775114915nt_nat(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ip(A4),P2))) = groups1707563613775114915nt_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_qz(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7130_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_nat,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups861055069439313189ex_nat(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_it(A4),P2))) = groups861055069439313189ex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_ra(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7131_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_int,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4694064378042380927al_int(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_il(A4),P2))) = groups4694064378042380927al_int(aa_fun6550077132417316957al_int(aTP_Lamm_rb(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7132_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_int,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups858564598930262913ex_int(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_it(A4),P2))) = groups858564598930262913ex_int(aa_fun4860376958843682909ex_int(aTP_Lamm_rc(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7133_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_real,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_il(A4),P2))) = groups1681761925125756287l_real(aa_fun6348708079608492509l_real(aTP_Lamm_rd(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7134_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_real,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_in(A4),P2))) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1750223370215434333t_real(aTP_Lamm_re(G),P2)),A4) ) ) ).

% prod.inter_filter
tff(fact_7135_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_real,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ip(A4),P2))) = groups2316167850115554303t_real(aa_fun6974987029372589917t_real(aTP_Lamm_rf(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7136_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_real,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_it(A4),P2))) = groups766887009212190081x_real(aa_fun2902006202505214429x_real(aTP_Lamm_rg(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7137_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_complex,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups713298508707869441omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_il(A4),P2))) = groups713298508707869441omplex(aa_fun6813637976755515871omplex(aTP_Lamm_rh(G),P2),A4) ) ) ).

% prod.inter_filter
tff(fact_7138_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M2: nat,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_la(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7139_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M2: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7140_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M2: nat,K2: nat,N: 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,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ri(G),K2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7141_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M2: nat,K2: nat,N: 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,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(G),K2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7142_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),one_one_real)) ) ).

% prod_le_1
tff(fact_7143_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),one_one_real)) ) ).

% prod_le_1
tff(fact_7144_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),one_one_real)) ) ).

% prod_le_1
tff(fact_7145_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7146_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7147_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7148_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A4)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7149_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A4))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A4)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7150_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A4))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),one_one_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A4)),one_one_int)) ) ).

% prod_le_1
tff(fact_7151_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A4))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),one_one_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_nat_int(groups705719431365010083at_int(F),A4)),one_one_int)) ) ).

% prod_le_1
tff(fact_7152_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X22))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups6464643781859351333omplex(H,S2)),groups6464643781859351333omplex(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7153_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X22))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_int_complex(H,X4)),aa_int_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups7440179247065528705omplex(H,S2)),groups7440179247065528705omplex(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7154_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X15),X22))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(H,X4)),aa_complex_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups3708469109370488835omplex(H,S2)),groups3708469109370488835omplex(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7155_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_nat_real(H,X4)),aa_nat_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_set_nat_real(groups129246275422532515t_real(H),S2)),aa_set_nat_real(groups129246275422532515t_real(G),S2))) ) ) ) ) ).

% prod.related
tff(fact_7156_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups2316167850115554303t_real(H,S2)),groups2316167850115554303t_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7157_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X15),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups766887009212190081x_real(H,S2)),groups766887009212190081x_real(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7158_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups73079841787564623at_rat(H,S2)),groups73079841787564623at_rat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7159_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_int_rat(H,X4)),aa_int_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups1072433553688619179nt_rat(H,S2)),groups1072433553688619179nt_rat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7160_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X15),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X15),Y15)),aa_rat_rat(times_times_rat(X22),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups225925009352817453ex_rat(H,S2)),groups225925009352817453ex_rat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7161_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,one_one_nat),one_one_nat))
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X15),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups1707563613775114915nt_nat(H,S2)),groups1707563613775114915nt_nat(G,S2))) ) ) ) ) ).

% prod.related
tff(fact_7162_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups713298508707869441omplex(G,insert_real(X,A4)) = groups713298508707869441omplex(G,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups713298508707869441omplex(G,insert_real(X,A4)) = aa_complex_complex(times_times_complex(aa_real_complex(G,X)),groups713298508707869441omplex(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7163_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => ( groups6464643781859351333omplex(G,insert_nat(X,A4)) = groups6464643781859351333omplex(G,A4) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => ( groups6464643781859351333omplex(G,insert_nat(X,A4)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,X)),groups6464643781859351333omplex(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7164_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( groups7440179247065528705omplex(G,insert_int(X,A4)) = groups7440179247065528705omplex(G,A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( groups7440179247065528705omplex(G,insert_int(X,A4)) = aa_complex_complex(times_times_complex(aa_int_complex(G,X)),groups7440179247065528705omplex(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7165_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups3708469109370488835omplex(G,insert_complex(X,A4)) = groups3708469109370488835omplex(G,A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups3708469109370488835omplex(G,insert_complex(X,A4)) = aa_complex_complex(times_times_complex(aa_complex_complex(G,X)),groups3708469109370488835omplex(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7166_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1681761925125756287l_real(G,insert_real(X,A4)) = groups1681761925125756287l_real(G,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1681761925125756287l_real(G,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7167_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X,A4)) = aa_set_nat_real(groups129246275422532515t_real(G),A4) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),aa_set_nat_real(groups129246275422532515t_real(G),A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7168_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( groups2316167850115554303t_real(G,insert_int(X,A4)) = groups2316167850115554303t_real(G,A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( groups2316167850115554303t_real(G,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7169_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups766887009212190081x_real(G,insert_complex(X,A4)) = groups766887009212190081x_real(G,A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups766887009212190081x_real(G,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7170_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups4061424788464935467al_rat(G,insert_real(X,A4)) = groups4061424788464935467al_rat(G,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups4061424788464935467al_rat(G,insert_real(X,A4)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X)),groups4061424788464935467al_rat(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7171_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => ( groups73079841787564623at_rat(G,insert_nat(X,A4)) = groups73079841787564623at_rat(G,A4) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => ( groups73079841787564623at_rat(G,insert_nat(X,A4)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_7172_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A3)),aa_real_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups4696554848551431203al_nat(F,A4)),groups4696554848551431203al_nat(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7173_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A3)),aa_complex_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7174_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A4))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A3)),aa_real_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7175_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A4))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A3)),aa_complex_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7176_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A3)),aa_int_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A4)),groups1707563613775114915nt_nat(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7177_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A4))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(F,A3)),aa_nat_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(F),A4)),aa_set_nat_int(groups705719431365010083at_int(G),B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7178_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A4))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(F,A3)),aa_int_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A4)),groups1705073143266064639nt_int(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7179_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(F,A3)),aa_nat_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)),aa_set_nat_nat(groups708209901874060359at_nat(G),B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7180_prod__dvd__prod__subset2,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,F: fun_set_nat_nat,G: fun_set_nat_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B4))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B4))
       => ( ! [A3: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(A3),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(F,A3)),aa_set_nat_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups4248547760180025341at_nat(F,A4)),groups4248547760180025341at_nat(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7181_prod__dvd__prod__subset2,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,F: fun_set_nat_int,G: fun_set_nat_int] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B4))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A4),B4))
       => ( ! [A3: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(A3),A4))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_set_nat_int(F,A3)),aa_set_nat_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups4246057289670975065at_int(F,A4)),groups4246057289670975065at_int(G,B4))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7182_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(F,B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7183_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups4077766827762148844at_nat(F,A4)),groups4077766827762148844at_nat(F,B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7184_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(F,B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7185_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B4))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A4),B4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups4075276357253098568at_int(F,A4)),groups4075276357253098568at_int(F,B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7186_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A4)),groups1707563613775114915nt_nat(F,B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7187_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(F),A4)),aa_set_nat_int(groups705719431365010083at_int(F),B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7188_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A4)),groups1705073143266064639nt_int(F,B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7189_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)),aa_set_nat_nat(groups708209901874060359at_nat(F),B4))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7190_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I2: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I2,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_real_real(J,aa_real_real(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_nat(H,aa_real_real(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups4696554848551431203al_nat(G,S2) = groups4696554848551431203al_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7191_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S2: set_real,I2: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_nat,H: fun_int_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_int_real(I2,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T2,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                 => ( aa_real_int(J,aa_int_real(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_int_nat(H,aa_real_int(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups4696554848551431203al_nat(G,S2) = groups1707563613775114915nt_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7192_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_complex,S2: set_real,I2: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_nat,H: fun_complex_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_complex_real(I2,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T2,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                 => ( aa_real_complex(J,aa_complex_real(I2,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_complex_nat(H,aa_real_complex(J,A3)) = aa_real_nat(G,A3) ) )
                     => ( groups4696554848551431203al_nat(G,S2) = groups861055069439313189ex_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7193_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_real,S2: set_int,I2: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_nat,H: fun_real_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_real_int(I2,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_int_real(J,aa_real_int(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_real_nat(H,aa_int_real(J,A3)) = aa_int_nat(G,A3) ) )
                     => ( groups1707563613775114915nt_nat(G,S2) = groups4696554848551431203al_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7194_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_int,S2: set_int,I2: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_int_int(I2,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T2,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                 => ( aa_int_int(J,aa_int_int(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_int_nat(H,aa_int_int(J,A3)) = aa_int_nat(G,A3) ) )
                     => ( groups1707563613775114915nt_nat(G,S2) = groups1707563613775114915nt_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7195_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_complex,S2: set_int,I2: fun_complex_int,J: fun_int_complex,T2: set_complex,G: fun_int_nat,H: fun_complex_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
             => ( aa_complex_int(I2,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T2,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                 => ( aa_int_complex(J,aa_complex_int(I2,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I2,B3)),minus_minus_set_int(S2,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_complex_nat(H,aa_int_complex(J,A3)) = aa_int_nat(G,A3) ) )
                     => ( groups1707563613775114915nt_nat(G,S2) = groups861055069439313189ex_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7196_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_real,S2: set_complex,I2: fun_real_complex,J: fun_complex_real,T2: set_real,G: fun_complex_nat,H: fun_real_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_real_complex(I2,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_complex_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_complex_real(J,aa_real_complex(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I2,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_real_nat(H,aa_complex_real(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups861055069439313189ex_nat(G,S2) = groups4696554848551431203al_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7197_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_int,S2: set_complex,I2: fun_int_complex,J: fun_complex_int,T2: set_int,G: fun_complex_nat,H: fun_int_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_int_complex(I2,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J,A3)),minus_minus_set_int(T2,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                 => ( aa_complex_int(J,aa_int_complex(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T2,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_int_complex(I2,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_int_nat(H,aa_complex_int(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups861055069439313189ex_nat(G,S2) = groups1707563613775114915nt_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7198_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_complex,S2: set_complex,I2: fun_complex_complex,J: fun_complex_complex,T2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
             => ( aa_complex_complex(I2,aa_complex_complex(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S2,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_complex_complex(J,A3)),minus_811609699411566653omplex(T2,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                 => ( aa_complex_complex(J,aa_complex_complex(I2,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T2,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_complex_complex(I2,B3)),minus_811609699411566653omplex(S2,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_nat(G,A3) = one_one_nat ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_nat(H,B3) = one_one_nat ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S2))
                         => ( aa_complex_nat(H,aa_complex_complex(J,A3)) = aa_complex_nat(G,A3) ) )
                     => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7199_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S2: set_real,I2: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
             => ( aa_real_real(I2,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                 => ( aa_real_real(J,aa_real_real(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T2,T3)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I2,B3)),minus_minus_set_real(S2,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_int(G,A3) = one_one_int ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_int(H,B3) = one_one_int ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_real_int(H,aa_real_real(J,A3)) = aa_real_int(G,A3) ) )
                     => ( groups4694064378042380927al_int(G,S2) = groups4694064378042380927al_int(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7200_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_7201_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_7202_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K2),L)),K2)) ) ).

% and_less_eq
tff(fact_7203_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4696554848551431203al_nat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_rj(G)))) = groups4696554848551431203al_nat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7204_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1707563613775114915nt_nat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_rk(G)))) = groups1707563613775114915nt_nat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7205_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rl(G)))) = groups861055069439313189ex_nat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7206_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4694064378042380927al_int(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_rm(G)))) = groups4694064378042380927al_int(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7207_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups858564598930262913ex_int(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rn(G)))) = groups858564598930262913ex_int(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7208_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1681761925125756287l_real(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_ro(G)))) = groups1681761925125756287l_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7209_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_rp(G)))) = groups2316167850115554303t_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7210_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rq(G)))) = groups766887009212190081x_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7211_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_rr(G)))) = groups713298508707869441omplex(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7212_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups7440179247065528705omplex(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_rs(G)))) = groups7440179247065528705omplex(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7213_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_rt(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_7214_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ni(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_7215_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N: nat,M2: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ru(G),N),M2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) ).

% prod.atLeastAtMost_rev
tff(fact_7216_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ky(G),N),M2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2)) ).

% prod.atLeastAtMost_rev
tff(fact_7217_less__1__prod2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I2),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7218_less__1__prod2,axiom,
    ! [I6: set_nat,I2: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I2),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I2)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7219_less__1__prod2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I2),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7220_less__1__prod2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I2),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7221_less__1__prod2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7222_less__1__prod2,axiom,
    ! [I6: set_nat,I2: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I2)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7223_less__1__prod2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7224_less__1__prod2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I2),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7225_less__1__prod2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I2),I6))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(F,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,I3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7226_less__1__prod2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I2),I6))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_complex_int(F,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_complex_int(F,I3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7227_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7228_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7229_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I6))) ) ) ) ).

% less_1_prod
tff(fact_7230_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7231_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7232_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7233_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7234_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7235_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_complex_int(F,I3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7236_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(F,I3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7237_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups3708469109370488835omplex(G,A4) = aa_complex_complex(times_times_complex(groups3708469109370488835omplex(G,minus_811609699411566653omplex(A4,B4))),groups3708469109370488835omplex(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7238_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( groups6464643781859351333omplex(G,A4) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,minus_minus_set_nat(A4,B4))),groups6464643781859351333omplex(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7239_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups766887009212190081x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,B4))),groups766887009212190081x_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7240_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A4,B4))),aa_set_nat_real(groups129246275422532515t_real(G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7241_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups225925009352817453ex_rat(G,A4) = aa_rat_rat(times_times_rat(groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups225925009352817453ex_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7242_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B4),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( groups73079841787564623at_rat(G,A4) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,minus_minus_set_nat(A4,B4))),groups73079841787564623at_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7243_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups861055069439313189ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7244_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups858564598930262913ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A4,B4))),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7245_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups7440179247065528705omplex(G,A4) = aa_complex_complex(times_times_complex(groups7440179247065528705omplex(G,minus_minus_set_int(A4,B4))),groups7440179247065528705omplex(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7246_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups2316167850115554303t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G,minus_minus_set_int(A4,B4))),groups2316167850115554303t_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7247_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,A4) = groups4696554848551431203al_nat(H,B4) )
              <=> ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7248_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,A4) = groups861055069439313189ex_nat(H,B4) )
              <=> ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7249_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,A4) = groups4694064378042380927al_int(H,B4) )
              <=> ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7250_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_int(G,A3) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,A4) = groups858564598930262913ex_int(H,B4) )
              <=> ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7251_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) )
              <=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7252_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) )
              <=> ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7253_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) )
              <=> ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7254_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) )
              <=> ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7255_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) )
              <=> ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7256_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,A4) = groups225925009352817453ex_rat(H,B4) )
              <=> ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7257_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_nat(H,B3) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,C4) = groups4696554848551431203al_nat(H,C4) )
               => ( groups4696554848551431203al_nat(G,A4) = groups4696554848551431203al_nat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7258_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_nat(H,B3) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,C4) = groups861055069439313189ex_nat(H,C4) )
               => ( groups861055069439313189ex_nat(G,A4) = groups861055069439313189ex_nat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7259_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_int(G,A3) = one_one_int ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_int(H,B3) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,C4) = groups4694064378042380927al_int(H,C4) )
               => ( groups4694064378042380927al_int(G,A4) = groups4694064378042380927al_int(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7260_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_int(G,A3) = one_one_int ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_int(H,B3) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,C4) = groups858564598930262913ex_int(H,C4) )
               => ( groups858564598930262913ex_int(G,A4) = groups858564598930262913ex_int(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7261_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
               => ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7262_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) )
               => ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7263_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) )
               => ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7264_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) )
               => ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7265_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) )
               => ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7266_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) )
               => ( groups225925009352817453ex_rat(G,A4) = groups225925009352817453ex_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7267_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7268_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7269_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7270_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7271_prod_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7272_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),S2) = aa_set_nat_real(groups129246275422532515t_real(G),T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7273_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,S2) = groups6464643781859351333omplex(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7274_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,S2) = groups73079841787564623at_rat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7275_prod_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,S2) = groups1707563613775114915nt_nat(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7276_prod_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,S2) = groups2316167850115554303t_real(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7277_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T2) = groups861055069439313189ex_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7278_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T2) = groups858564598930262913ex_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7279_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T2) = groups766887009212190081x_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7280_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T2) = groups3708469109370488835omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7281_prod_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T2) = groups225925009352817453ex_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7282_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),T2) = aa_set_nat_real(groups129246275422532515t_real(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7283_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,T2) = groups6464643781859351333omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7284_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T2,S2)))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,T2) = groups73079841787564623at_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7285_prod_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,T2) = groups1707563613775114915nt_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7286_prod_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T2,S2)))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,T2) = groups2316167850115554303t_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7287_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups4696554848551431203al_nat(G,S2) = groups4696554848551431203al_nat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7288_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7289_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_int(H,X4) = one_one_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups4694064378042380927al_int(G,S2) = groups4694064378042380927al_int(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7290_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(H,X4) = one_one_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7291_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_real(H,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7292_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(H,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7293_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7294_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7295_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups4061424788464935467al_rat(G,S2) = groups4061424788464935467al_rat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7296_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7297_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups4696554848551431203al_nat(G,T2) = groups4696554848551431203al_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7298_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_nat(G,X4) = aa_complex_nat(H,X4) ) )
           => ( groups861055069439313189ex_nat(G,T2) = groups861055069439313189ex_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7299_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_int(G,X4) = one_one_int ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_int(G,X4) = aa_real_int(H,X4) ) )
           => ( groups4694064378042380927al_int(G,T2) = groups4694064378042380927al_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7300_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_int(G,X4) = aa_complex_int(H,X4) ) )
           => ( groups858564598930262913ex_int(G,T2) = groups858564598930262913ex_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7301_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_real(G,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,T2) = groups1681761925125756287l_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7302_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,T2) = groups766887009212190081x_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7303_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups713298508707869441omplex(G,T2) = groups713298508707869441omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7304_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,T2) = groups3708469109370488835omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7305_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T2,S2)))
             => ( aa_real_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups4061424788464935467al_rat(G,T2) = groups4061424788464935467al_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7306_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T2,S2)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups225925009352817453ex_rat(G,T2) = groups225925009352817453ex_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7307_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7308_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: 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,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7309_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7310_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: 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,N))) = 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),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7311_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: 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,N))) = 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),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7312_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,M2)),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7313_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M2)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7314_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M2)),groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7315_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M2)),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7316_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M2)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7317_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,aa_nat_nat(suc,N))),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7318_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7319_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7320_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N))),aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7321_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7322_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_7323_log__of__power__eq,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M2)) ) ) ) ).

% log_of_power_eq
tff(fact_7324_less__log__of__power,axiom,
    ! [B: real,N: nat,M2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)),M2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M2))) ) ) ).

% less_log_of_power
tff(fact_7325_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% pi_less_4
tff(fact_7326_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_rv(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7327_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7328_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7329_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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_la(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7330_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = 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_kq(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7331_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,M2)),groups6464643781859351333omplex(aTP_Lamm_rv(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7332_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M2)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7333_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M2)),groups73079841787564623at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7334_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( 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,M2),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M2)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_la(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7335_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( 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,M2),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M2)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7336_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N: 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)),N)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_la(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_7337_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: 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)),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_7338_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,I3)),aa_complex_real(G,I3))) ) )
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7339_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,I3)),aa_real_real(G,I3))) ) )
       => ( ( A4 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7340_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,I3)),aa_nat_real(G,I3))) ) )
       => ( ( A4 != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),aa_set_nat_real(groups129246275422532515t_real(G),A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7341_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,I3)),aa_int_real(G,I3))) ) )
       => ( ( A4 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7342_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3))) ) )
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7343_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) ) )
       => ( ( A4 != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7344_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) ) )
       => ( ( A4 != bot_bot_set_nat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7345_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) ) )
       => ( ( A4 != bot_bot_set_int )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7346_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3))) ) )
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7347_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) ) )
       => ( ( A4 != bot_bot_set_real )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4696554848551431203al_nat(F,A4)),groups4696554848551431203al_nat(G,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_7348_even__prod__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups1707563613775114915nt_nat(F,A4)))
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7349_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups861055069439313189ex_nat(F,A4)))
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_complex_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7350_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups4077766827762148844at_nat(F,A4)))
      <=> ? [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_Pro4243173770044397973at_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7351_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups858564598930262913ex_int(F,A4)))
      <=> ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_complex_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7352_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups4075276357253098568at_int(F,A4)))
      <=> ? [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Pro4240683299535347697at_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7353_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups705719431365010083at_int(F),A4)))
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7354_even__prod__iff,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups1705073143266064639nt_int(F,A4)))
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7355_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups708209901874060359at_nat(F),A4)))
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(F,X5))) ) ) ) ).

% even_prod_iff
tff(fact_7356_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups3708469109370488835omplex(G,insert_complex(X,A4)) = aa_complex_complex(times_times_complex(aa_complex_complex(G,X)),groups3708469109370488835omplex(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_7357_prod_Oinsert__remove,axiom,
    ! [A4: set_real,G: fun_real_complex,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups713298508707869441omplex(G,insert_real(X,A4)) = aa_complex_complex(times_times_complex(aa_real_complex(G,X)),groups713298508707869441omplex(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_7358_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_complex,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups7440179247065528705omplex(G,insert_int(X,A4)) = aa_complex_complex(times_times_complex(aa_int_complex(G,X)),groups7440179247065528705omplex(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_7359_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G: fun_nat_complex,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups6464643781859351333omplex(G,insert_nat(X,A4)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,X)),groups6464643781859351333omplex(G,minus_minus_set_nat(A4,insert_nat(X,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_7360_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_real,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups766887009212190081x_real(G,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_7361_prod_Oinsert__remove,axiom,
    ! [A4: set_real,G: fun_real_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1681761925125756287l_real(G,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_7362_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G: fun_int_real,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups2316167850115554303t_real(G,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_7363_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G: fun_nat_real,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A4,insert_nat(X,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_7364_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_rat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups225925009352817453ex_rat(G,insert_complex(X,A4)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_7365_prod_Oinsert__remove,axiom,
    ! [A4: set_real,G: fun_real_rat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4061424788464935467al_rat(G,insert_real(X,A4)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X)),groups4061424788464935467al_rat(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_7366_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups3708469109370488835omplex(G,A4) = aa_complex_complex(times_times_complex(aa_complex_complex(G,X)),groups3708469109370488835omplex(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_7367_prod_Oremove,axiom,
    ! [A4: set_real,X: real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups713298508707869441omplex(G,A4) = aa_complex_complex(times_times_complex(aa_real_complex(G,X)),groups713298508707869441omplex(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_7368_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( groups7440179247065528705omplex(G,A4) = aa_complex_complex(times_times_complex(aa_int_complex(G,X)),groups7440179247065528705omplex(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_7369_prod_Oremove,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(X),A4))
       => ( groups6464643781859351333omplex(G,A4) = aa_complex_complex(times_times_complex(aa_nat_complex(G,X)),groups6464643781859351333omplex(G,minus_minus_set_nat(A4,insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_7370_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups766887009212190081x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_7371_prod_Oremove,axiom,
    ! [A4: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1681761925125756287l_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_7372_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( groups2316167850115554303t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_7373_prod_Oremove,axiom,
    ! [A4: set_nat,X: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A4,insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_7374_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups225925009352817453ex_rat(G,A4) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_7375_prod_Oremove,axiom,
    ! [A4: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups4061424788464935467al_rat(G,A4) = aa_rat_rat(times_times_rat(aa_real_rat(G,X)),groups4061424788464935467al_rat(G,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_7376_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_complex,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),groups6464643781859351333omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7377_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_real,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),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,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7378_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_rat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N))),groups73079841787564623at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7379_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_int,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = 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,M2),N))),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,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7380_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: fun_nat_nat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = 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,M2),N))),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,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_7381_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_7382_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_7383_le__log__of__power,axiom,
    ! [B: real,N: nat,M2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)),M2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M2))) ) ) ).

% le_log_of_power
tff(fact_7384_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_7385_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_nat_power
tff(fact_7386_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_complex,C2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aa_fun4268748527592579270omplex(aTP_Lamm_rw(A),B),C2),S2) = aa_complex_complex(times_times_complex(aa_complex_complex(B,A)),groups3708469109370488835omplex(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aa_fun4268748527592579270omplex(aTP_Lamm_rw(A),B),C2),S2) = groups3708469109370488835omplex(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7387_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex,C2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aa_fun2106233272836097472omplex(aTP_Lamm_rx(A),B),C2),S2) = aa_complex_complex(times_times_complex(aa_real_complex(B,A)),groups713298508707869441omplex(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aa_fun2106233272836097472omplex(aTP_Lamm_rx(A),B),C2),S2) = groups713298508707869441omplex(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7388_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex,C2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aa_fun8701580876069687872omplex(aTP_Lamm_ry(A),B),C2),S2) = aa_complex_complex(times_times_complex(aa_int_complex(B,A)),groups7440179247065528705omplex(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aa_fun8701580876069687872omplex(aTP_Lamm_ry(A),B),C2),S2) = groups7440179247065528705omplex(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7389_prod_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex,C2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rz(A),B),C2),S2) = aa_complex_complex(times_times_complex(aa_nat_complex(B,A)),groups6464643781859351333omplex(C2,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat)))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rz(A),B),C2),S2) = groups6464643781859351333omplex(C2,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7390_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_lh(A),B),C2),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(B,A)),groups766887009212190081x_real(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_lh(A),B),C2),S2) = groups766887009212190081x_real(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7391_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ll(A),B),C2),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(B,A)),groups1681761925125756287l_real(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ll(A),B),C2),S2) = groups1681761925125756287l_real(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7392_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_real,C2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lo(A),B),C2),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(B,A)),groups2316167850115554303t_real(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lo(A),B),C2),S2) = groups2316167850115554303t_real(C2,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7393_prod_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real,C2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sa(A),B),C2)),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(B,A)),aa_set_nat_real(groups129246275422532515t_real(C2),minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat)))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sa(A),B),C2)),S2) = aa_set_nat_real(groups129246275422532515t_real(C2),minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7394_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_li(A),B),C2),S2) = aa_rat_rat(times_times_rat(aa_complex_rat(B,A)),groups225925009352817453ex_rat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_li(A),B),C2),S2) = groups225925009352817453ex_rat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7395_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lm(A),B),C2),S2) = aa_rat_rat(times_times_rat(aa_real_rat(B,A)),groups4061424788464935467al_rat(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lm(A),B),C2),S2) = groups4061424788464935467al_rat(C2,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7396_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb3: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb3),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb3),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb3),Xa2))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb3),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_7397_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: nat,A: nat,F: fun_nat_fun_nat_nat,Acc2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = Acc2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),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_7398_pi__half__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_less_two
tff(fact_7399_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7400_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7401_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B4,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,B3))) )
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),A4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A4)),aa_set_nat_real(groups129246275422532515t_real(F),B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7402_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7403_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7404_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),B4))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B4,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B3))) )
         => ( ! [A3: nat] :
                ( pp(aa_set_nat_bool(member_nat(A3),A4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7405_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A4)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A4))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7406_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A4)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_complex_int(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7407_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7408_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B4))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_7409_prod__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( aa_complex_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F,A4)),aa_complex_complex(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups3708469109370488835omplex(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7410_prod__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_rat,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( aa_complex_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups225925009352817453ex_rat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_rat_rat(divide_divide_rat(groups225925009352817453ex_rat(F,A4)),aa_complex_rat(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups225925009352817453ex_rat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups225925009352817453ex_rat(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7411_prod__diff1,axiom,
    ! [A4: set_real,F: fun_real_complex,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( aa_real_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_real_bool(member_real(A),A4))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F,A4)),aa_real_complex(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A4))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups713298508707869441omplex(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7412_prod__diff1,axiom,
    ! [A4: set_real,F: fun_real_rat,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( aa_real_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_real_bool(member_real(A),A4))
           => ( groups4061424788464935467al_rat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_rat_rat(divide_divide_rat(groups4061424788464935467al_rat(F,A4)),aa_real_rat(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A4))
           => ( groups4061424788464935467al_rat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups4061424788464935467al_rat(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7413_prod__diff1,axiom,
    ! [A4: set_int,F: fun_int_complex,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( aa_int_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_int_bool(member_int(A),A4))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F,A4)),aa_int_complex(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A4))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups7440179247065528705omplex(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7414_prod__diff1,axiom,
    ! [A4: set_int,F: fun_int_rat,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( aa_int_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_int_bool(member_int(A),A4))
           => ( groups1072433553688619179nt_rat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_rat_rat(divide_divide_rat(groups1072433553688619179nt_rat(F,A4)),aa_int_rat(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A4))
           => ( groups1072433553688619179nt_rat(F,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups1072433553688619179nt_rat(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7415_prod__diff1,axiom,
    ! [A4: set_nat,F: fun_nat_complex,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_nat_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A4))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(F,A4)),aa_nat_complex(F,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A4))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = groups6464643781859351333omplex(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7416_prod__diff1,axiom,
    ! [A4: set_nat,F: fun_nat_rat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_nat_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A4))
           => ( groups73079841787564623at_rat(F,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(F,A4)),aa_nat_rat(F,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A4))
           => ( groups73079841787564623at_rat(F,minus_minus_set_nat(A4,insert_nat(A,bot_bot_set_nat))) = groups73079841787564623at_rat(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7417_prod__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_nat,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( aa_complex_nat(F,A) != zero_zero_nat )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups861055069439313189ex_nat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups861055069439313189ex_nat(F,A4)),aa_complex_nat(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups861055069439313189ex_nat(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups861055069439313189ex_nat(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7418_prod__diff1,axiom,
    ! [A4: set_real,F: fun_real_nat,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( aa_real_nat(F,A) != zero_zero_nat )
       => ( ( pp(aa_set_real_bool(member_real(A),A4))
           => ( groups4696554848551431203al_nat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4696554848551431203al_nat(F,A4)),aa_real_nat(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A4))
           => ( groups4696554848551431203al_nat(F,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups4696554848551431203al_nat(F,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_7419_log__of__power__less,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_7420_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( ( B != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_7421_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aTP_Lamm_sb(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_7422_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_sc,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_7423_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_sd(A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_7424_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_se,A)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_7425_pi__half__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_gt_zero
tff(fact_7426_pochhammer__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_sf(A),N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7427_pochhammer__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_sg(A),N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7428_pochhammer__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sh(A),N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7429_pochhammer__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_si(A),N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7430_m2pi__less__pi,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_7431_log__of__power__le,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_7432_prod_Oin__pairs,axiom,
    ! [G: fun_nat_complex,M2: nat,N: 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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups6464643781859351333omplex(aTP_Lamm_sj(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.in_pairs
tff(fact_7433_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M2: nat,N: 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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_sk(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.in_pairs
tff(fact_7434_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M2: nat,N: nat] : groups73079841787564623at_rat(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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_sl(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.in_pairs
tff(fact_7435_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M2: nat,N: 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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_sm(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.in_pairs
tff(fact_7436_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M2: nat,N: 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,aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_sn(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) ).

% prod.in_pairs
tff(fact_7437_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(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_so(F),A),B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_7438_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_sp(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7439_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(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_sq(F),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7440_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_sr(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7441_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_ss(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_7442_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_sf(A),N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7443_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_sg(A),N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7444_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sh(A),N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7445_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_si(A),N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7446_minus__pi__half__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_7447_less__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2)))) ) ).

% less_log2_of_power
tff(fact_7448_le__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2)))) ) ).

% le_log2_of_power
tff(fact_7449_log2__of__power__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_7450_log2__of__power__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_7451_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_7452_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_7453_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K2)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_7454_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_st,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_7455_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => pp(aa_complex_bool(sums_complex(aTP_Lamm_su(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_7456_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_7457_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa2)))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                  & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa2))) ) ) ) ).

% and_int.pelims
tff(fact_7458_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K2,L)))
     => ( ( ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K2),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_7459_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_7460_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W2),Z) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_7461_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_7462_sums__zero,axiom,
    pp(aa_complex_bool(sums_complex(aTP_Lamm_ok),zero_zero_complex)) ).

% sums_zero
tff(fact_7463_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_fr),zero_zero_real)) ).

% sums_zero
tff(fact_7464_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_bz),zero_zero_nat)) ).

% sums_zero
tff(fact_7465_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_ol),zero_zero_int)) ).

% sums_zero
tff(fact_7466_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = one_one_real ) ) ) ).

% powr_zero_eq_one
tff(fact_7467_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_7468_floor__zero,axiom,
    aa_rat_int(archim3151403230148437115or_rat,zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_7469_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)))
    <=> ( X != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_7470_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel_iff
tff(fact_7471_prod__eq__1__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups1707563613775114915nt_nat(F,A4) = one_one_nat )
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
           => ( aa_int_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7472_prod__eq__1__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups861055069439313189ex_nat(F,A4) = one_one_nat )
      <=> ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
           => ( aa_complex_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7473_prod__eq__1__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( ( groups4077766827762148844at_nat(F,A4) = one_one_nat )
      <=> ! [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
           => ( aa_Pro4243173770044397973at_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7474_prod__eq__1__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A4) = one_one_nat )
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
           => ( aa_nat_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7475_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_7476_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_7477_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_7478_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% powr_le_cancel_iff
tff(fact_7479_prod__pos__nat__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A4)))
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7480_prod__pos__nat__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A4)))
      <=> ! [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7481_prod__pos__nat__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4077766827762148844at_nat(F,A4)))
      <=> ! [X5: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7482_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A4)))
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7483_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_7484_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_7485_zero__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_floor
tff(fact_7486_zero__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X)) ) ).

% zero_le_floor
tff(fact_7487_floor__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% floor_less_zero
tff(fact_7488_floor__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat)) ) ).

% floor_less_zero
tff(fact_7489_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_le_floor
tff(fact_7490_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_le_floor
tff(fact_7491_zero__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% zero_less_floor
tff(fact_7492_zero__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% zero_less_floor
tff(fact_7493_floor__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_le_zero
tff(fact_7494_floor__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_le_zero
tff(fact_7495_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_numeral
tff(fact_7496_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_numeral
tff(fact_7497_one__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% one_le_floor
tff(fact_7498_one__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% one_le_floor
tff(fact_7499_floor__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_less_one
tff(fact_7500_floor__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_less_one
tff(fact_7501_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_7502_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_7503_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_jr(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_7504_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_jt(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_7505_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_7506_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_7507_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X)) ) ).

% numeral_less_floor
tff(fact_7508_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_less_floor
tff(fact_7509_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% floor_le_numeral
tff(fact_7510_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% floor_le_numeral
tff(fact_7511_one__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_7512_one__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_7513_floor__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_7514_floor__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_7515_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_7516_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_7517_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% floor_less_neg_numeral
tff(fact_7518_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% floor_less_neg_numeral
tff(fact_7519_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_less_floor
tff(fact_7520_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_less_floor
tff(fact_7521_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% floor_le_neg_numeral
tff(fact_7522_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% floor_le_neg_numeral
tff(fact_7523_int__prod,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A4)) = groups1705073143266064639nt_int(aTP_Lamm_mc(F),A4) ).

% int_prod
tff(fact_7524_int__prod,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A4)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_me(F)),A4) ).

% int_prod
tff(fact_7525_sums__0,axiom,
    ! [F: fun_nat_complex] :
      ( ! [N3: nat] : aa_nat_complex(F,N3) = zero_zero_complex
     => pp(aa_complex_bool(sums_complex(F),zero_zero_complex)) ) ).

% sums_0
tff(fact_7526_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real
     => pp(aa_real_bool(sums_real(F),zero_zero_real)) ) ).

% sums_0
tff(fact_7527_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F),zero_zero_nat)) ) ).

% sums_0
tff(fact_7528_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int
     => pp(aa_int_bool(sums_int(F),zero_zero_int)) ) ).

% sums_0
tff(fact_7529_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,T: real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(G,N3)))
     => ( pp(aa_real_bool(sums_real(F),S))
       => ( pp(aa_real_bool(sums_real(G),T))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,S),T)) ) ) ) ).

% sums_le
tff(fact_7530_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(G,N3)))
     => ( pp(aa_nat_bool(sums_nat(F),S))
       => ( pp(aa_nat_bool(sums_nat(G),T))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T)) ) ) ) ).

% sums_le
tff(fact_7531_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,T: int] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(G,N3)))
     => ( pp(aa_int_bool(sums_int(F),S))
       => ( pp(aa_int_bool(sums_int(G),T))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S),T)) ) ) ) ).

% sums_le
tff(fact_7532_sums__single,axiom,
    ! [I2: nat,F: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ft(I2),F)),aa_nat_complex(F,I2))) ).

% sums_single
tff(fact_7533_sums__single,axiom,
    ! [I2: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_sv(I2),F)),aa_nat_real(F,I2))) ).

% sums_single
tff(fact_7534_sums__single,axiom,
    ! [I2: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sw(I2),F)),aa_nat_nat(F,I2))) ).

% sums_single
tff(fact_7535_sums__single,axiom,
    ! [I2: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_sx(I2),F)),aa_nat_int(F,I2))) ).

% sums_single
tff(fact_7536_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( pp(aa_nat_bool(sums_nat(F),A))
     => ( pp(aa_nat_bool(sums_nat(G),B))
       => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gu(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% sums_add
tff(fact_7537_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( pp(aa_int_bool(sums_int(F),A))
     => ( pp(aa_int_bool(sums_int(G),B))
       => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_sy(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% sums_add
tff(fact_7538_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => ( pp(aa_real_bool(sums_real(G),B))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_gw(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% sums_add
tff(fact_7539_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))) ) ).

% floor_mono
tff(fact_7540_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))) ) ).

% floor_mono
tff(fact_7541_of__int__floor__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X)) ).

% of_int_floor_le
tff(fact_7542_of__int__floor__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X)) ).

% of_int_floor_le
tff(fact_7543_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% floor_less_cancel
tff(fact_7544_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% floor_less_cancel
tff(fact_7545_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_7546_powr__non__neg,axiom,
    ! [A: real,X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real)) ).

% powr_non_neg
tff(fact_7547_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel
tff(fact_7548_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))) ) ) ).

% powr_less_mono
tff(fact_7549_sums__mult__iff,axiom,
    ! [C2: complex,F: fun_nat_complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sz(C2),F)),aa_complex_complex(times_times_complex(C2),D)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult_iff
tff(fact_7550_sums__mult__iff,axiom,
    ! [C2: real,F: fun_nat_real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ta(C2),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),D)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult_iff
tff(fact_7551_sums__mult2__iff,axiom,
    ! [C2: complex,F: fun_nat_complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tb(C2),F)),aa_complex_complex(times_times_complex(D),C2)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_7552_sums__mult2__iff,axiom,
    ! [C2: real,F: fun_nat_real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_tc(C2),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C2)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_7553_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% le_floor_iff
tff(fact_7554_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% le_floor_iff
tff(fact_7555_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% floor_less_iff
tff(fact_7556_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% floor_less_iff
tff(fact_7557_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_mono2'
tff(fact_7558_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).

% powr_less_mono2
tff(fact_7559_le__floor__add,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))) ).

% le_floor_add
tff(fact_7560_le__floor__add,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y)))) ).

% le_floor_add
tff(fact_7561_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_7562_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ) ) ).

% gr_one_powr
tff(fact_7563_int__add__floor,axiom,
    ! [Z: int,X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,X)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),X)) ).

% int_add_floor
tff(fact_7564_int__add__floor,axiom,
    ! [Z: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),X)) ).

% int_add_floor
tff(fact_7565_floor__add__int,axiom,
    ! [X: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_7566_floor__add__int,axiom,
    ! [X: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_7567_sums__mult__D,axiom,
    ! [C2: complex,F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sz(C2),F)),A))
     => ( ( C2 != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C2))) ) ) ).

% sums_mult_D
tff(fact_7568_sums__mult__D,axiom,
    ! [C2: real,F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ta(C2),F)),A))
     => ( ( C2 != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C2))) ) ) ).

% sums_mult_D
tff(fact_7569_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aTP_Lamm_rv(F)),S))
       => pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_7570_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_kr(F)),S))
       => pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_7571_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_kr(F)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_7572_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_kq(F)),L))
     => pp(aa_nat_bool(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_7573_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_la(F)),L))
     => pp(aa_int_bool(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_7574_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_kr(F)),L))
     => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_7575_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_complex,S: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ( aa_nat_complex(F,I3) = zero_zero_complex ) )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_td(N),F)),S))
      <=> pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_7576_sums__zero__iff__shift,axiom,
    ! [N: nat,F: fun_nat_real,S: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ( aa_nat_real(F,I3) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_te(N),F)),S))
      <=> pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_7577_powr__add,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)) ).

% powr_add
tff(fact_7578_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_complex(F,N3) = zero_zero_complex ) )
       => pp(aa_complex_bool(sums_complex(F),groups2073611262835488442omplex(F,N4))) ) ) ).

% sums_finite
tff(fact_7579_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_int(F,N3) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),aa_set_nat_int(groups3539618377306564664at_int(F),N4))) ) ) ).

% sums_finite
tff(fact_7580_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_nat(F,N3) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N4))) ) ) ).

% sums_finite
tff(fact_7581_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N4))
           => ( aa_nat_real(F,N3) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N4))) ) ) ).

% sums_finite
tff(fact_7582_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tf(P2),F)),groups2073611262835488442omplex(F,collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_7583_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_tg(P2),F)),aa_set_nat_int(groups3539618377306564664at_int(F),collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_7584_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_th(P2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_7585_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ti(P2),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_7586_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tj(A4),F)),groups2073611262835488442omplex(F,A4))) ) ).

% sums_If_finite_set
tff(fact_7587_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_tk(A4),F)),aa_set_nat_int(groups3539618377306564664at_int(F),A4))) ) ).

% sums_If_finite_set
tff(fact_7588_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tl(A4),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4))) ) ).

% sums_If_finite_set
tff(fact_7589_sums__If__finite__set,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_tm(A4),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A4))) ) ).

% sums_If_finite_set
tff(fact_7590_prod__int__eq,axiom,
    ! [I2: nat,J: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I2),J)) = groups1705073143266064639nt_int(aTP_Lamm_cf,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_7591_one__add__floor,axiom,
    ! [X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% one_add_floor
tff(fact_7592_one__add__floor,axiom,
    ! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) ).

% one_add_floor
tff(fact_7593_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = K2 )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K2))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_7594_powser__sums__if,axiom,
    ! [M2: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_tn(M2),Z)),aa_nat_complex(power_power_complex(Z),M2))) ).

% powser_sums_if
tff(fact_7595_powser__sums__if,axiom,
    ! [M2: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_to(M2),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M2))) ).

% powser_sums_if
tff(fact_7596_powser__sums__if,axiom,
    ! [M2: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_tp(M2),Z)),aa_nat_int(power_power_int(Z),M2))) ).

% powser_sums_if
tff(fact_7597_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_jr(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7598_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_jt(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7599_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N) ) ) ).

% powr_realpow
tff(fact_7600_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_less_iff
tff(fact_7601_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_7602_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_less_iff
tff(fact_7603_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_7604_sums__iff__shift,axiom,
    ! [F: fun_nat_real,N: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_kt(F),N)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_iff_shift
tff(fact_7605_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S: real,N: nat] :
      ( pp(aa_real_bool(sums_real(F),S))
     => pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_kt(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).

% sums_split_initial_segment
tff(fact_7606_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,N: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_kt(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N)))))
    <=> pp(aa_real_bool(sums_real(F),S)) ) ).

% sums_iff_shift'
tff(fact_7607_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq
tff(fact_7608_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_7609_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A4: set_nat,S5: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G),S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( ( S5 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_tq(G),F)),A4)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_tr(G),A4),F)),S5)) ) ) ) ).

% sums_If_finite_set'
tff(fact_7610_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_7611_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J))) = groups1705073143266064639nt_int(aTP_Lamm_cf,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)))) ).

% prod_int_plus_eq
tff(fact_7612_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Z ) ) ) ).

% floor_unique
tff(fact_7613_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z ) ) ) ).

% floor_unique
tff(fact_7614_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real))) ) ) ).

% floor_eq_iff
tff(fact_7615_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat))) ) ) ).

% floor_eq_iff
tff(fact_7616_floor__split,axiom,
    ! [P2: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P2,aa_real_int(archim6058952711729229775r_real,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I4)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real))) )
         => pp(aa_int_bool(P2,I4)) ) ) ).

% floor_split
tff(fact_7617_floor__split,axiom,
    ! [P2: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P2,aa_rat_int(archim3151403230148437115or_rat,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat))) )
         => pp(aa_int_bool(P2,I4)) ) ) ).

% floor_split
tff(fact_7618_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).

% le_mult_floor
tff(fact_7619_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor
tff(fact_7620_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).

% less_floor_iff
tff(fact_7621_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% less_floor_iff
tff(fact_7622_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).

% floor_le_iff
tff(fact_7623_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% floor_le_iff
tff(fact_7624_binomial__maximum_H,axiom,
    ! [N: nat,K2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_7625_binomial__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_mono
tff(fact_7626_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) ) ) ) ).

% binomial_antimono
tff(fact_7627_binomial__maximum,axiom,
    ! [N: nat,K2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_7628_floor__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_7629_floor__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_7630_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_7631_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_7632_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_le_iff
tff(fact_7633_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_7634_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_le_iff
tff(fact_7635_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq2
tff(fact_7636_ln__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups1681761925125756287l_real(F,I6)) = groups8097168146408367636l_real(aTP_Lamm_ts(F),I6) ) ) ) ).

% ln_prod
tff(fact_7637_ln__prod,axiom,
    ! [I6: set_set_nat,F: fun_set_nat_real] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,I6))
     => ( ! [I3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(I3),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_set_nat_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups3619160379726066777t_real(F,I6)) = groups5107569545109728110t_real(aTP_Lamm_tt(F),I6) ) ) ) ).

% ln_prod
tff(fact_7638_ln__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups2316167850115554303t_real(F,I6)) = groups8778361861064173332t_real(aTP_Lamm_tu(F),I6) ) ) ) ).

% ln_prod
tff(fact_7639_ln__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups766887009212190081x_real(F,I6)) = groups5808333547571424918x_real(aTP_Lamm_tv(F),I6) ) ) ) ).

% ln_prod
tff(fact_7640_ln__prod,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,I6))
     => ( ! [I3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(I3),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Pro6168199777849092593t_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups6036352826371341000t_real(F,I6)) = groups4567486121110086003t_real(aTP_Lamm_tw(F),I6) ) ) ) ).

% ln_prod
tff(fact_7641_ln__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,aa_set_nat_real(groups129246275422532515t_real(F),I6)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_tx(F)),I6) ) ) ) ).

% ln_prod
tff(fact_7642_floor__divide__lower,axiom,
    ! [Q2: real,P: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P),Q2)))),Q2)),P)) ) ).

% floor_divide_lower
tff(fact_7643_floor__divide__lower,axiom,
    ! [Q2: rat,P: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P),Q2)))),Q2)),P)) ) ).

% floor_divide_lower
tff(fact_7644_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K2)))) ) ).

% binomial_less_binomial_Suc
tff(fact_7645_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_strict_mono
tff(fact_7646_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) ) ) ) ).

% binomial_strict_antimono
tff(fact_7647_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),A))) ) ) ).

% ln_powr_bound
tff(fact_7648_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),X))) ) ) ).

% ln_powr_bound2
tff(fact_7649_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_7650_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_7651_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_7652_floor__divide__upper,axiom,
    ! [Q2: real,P: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,P),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P),Q2)))),one_one_real)),Q2))) ) ).

% floor_divide_upper
tff(fact_7653_floor__divide__upper,axiom,
    ! [Q2: rat,P: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P),aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P),Q2)))),one_one_rat)),Q2))) ) ).

% floor_divide_upper
tff(fact_7654_geometric__sums,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C2)),one_one_real))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(power_power_real,C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C2)))) ) ).

% geometric_sums
tff(fact_7655_geometric__sums,axiom,
    ! [C2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C2)),one_one_real))
     => pp(aa_complex_bool(sums_complex(power_power_complex(C2)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C2)))) ) ).

% geometric_sums
tff(fact_7656_round__def,axiom,
    ! [X: real] : aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_7657_round__def,axiom,
    ! [X: rat] : aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_7658_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_7659_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P2: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(A0,A1)))
     => ( ! [K: int,L3: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K,L3)))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P2,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P2,K),L3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P2,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_7660_and__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ( M2 = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M2),N) = zero_zero_nat ) )
      & ( ~ ( ( M2 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_7661_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_7662_and__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_7663_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_7664_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_7665_zero__less__binomial__iff,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N)) ) ).

% zero_less_binomial_iff
tff(fact_7666_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_7667_binomial__Suc__Suc,axiom,
    ! [N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K2))) ).

% binomial_Suc_Suc
tff(fact_7668_binomial__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% binomial_eq_0_iff
tff(fact_7669_binomial__0__Suc,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_7670_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_7671_binomial__eq__0,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_7672_binomial__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% binomial_symmetric
tff(fact_7673_choose__mult__lemma,axiom,
    ! [M2: nat,R2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),R2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),R2)),K2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),R2)),M2)) ).

% choose_mult_lemma
tff(fact_7674_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% binomial_le_pow
tff(fact_7675_zero__less__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) ) ).

% zero_less_binomial
tff(fact_7676_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_7677_choose__mult,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2))) ) ) ) ).

% choose_mult
tff(fact_7678_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_7679_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_7680_binomial__le__pow2,axiom,
    ! [N: nat,K2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_7681_choose__reduce__nat,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_7682_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_7683_binomial__addition__formula,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K2)) ) ) ).

% binomial_addition_formula
tff(fact_7684_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P2: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(A0,A1)))
     => ( ! [I3: int,J2: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I3,J2)))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P2,I3),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P2,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_7685_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_ty(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_7686_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_tz(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_7687_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ua,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_7688_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ub,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_7689_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_uc(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_7690_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_ud(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_7691_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ue,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_7692_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_uf,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_7693_arcosh__def,axiom,
    ! [X: real] : aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_7694_round__altdef,axiom,
    ! [X: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim7802044766580827645g_real,X) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,X) ) ) ) ).

% round_altdef
tff(fact_7695_round__altdef,axiom,
    ! [X: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim2889992004027027881ng_rat,X) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,X) ) ) ) ).

% round_altdef
tff(fact_7696_atMost__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_atMost_nat,X) = aa_nat_set_nat(set_ord_atMost_nat,Y) )
    <=> ( X = Y ) ) ).

% atMost_eq_iff
tff(fact_7697_atMost__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( set_ord_atMost_int(X) = set_ord_atMost_int(Y) )
    <=> ( X = Y ) ) ).

% atMost_eq_iff
tff(fact_7698_atMost__iff,axiom,
    ! [I2: real,K2: real] :
      ( pp(aa_set_real_bool(member_real(I2),set_ord_atMost_real(K2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),K2)) ) ).

% atMost_iff
tff(fact_7699_atMost__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I2),set_or4236626031148496127et_nat(K2)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I2),K2)) ) ).

% atMost_iff
tff(fact_7700_atMost__iff,axiom,
    ! [I2: set_int,K2: set_int] :
      ( member_set_int(I2,set_or58775011639299419et_int(K2))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(I2),K2)) ) ).

% atMost_iff
tff(fact_7701_atMost__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( pp(aa_set_rat_bool(member_rat(I2),set_ord_atMost_rat(K2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),K2)) ) ).

% atMost_iff
tff(fact_7702_atMost__iff,axiom,
    ! [I2: num,K2: num] :
      ( pp(aa_set_num_bool(member_num(I2),set_ord_atMost_num(K2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I2),K2)) ) ).

% atMost_iff
tff(fact_7703_atMost__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_set_nat_bool(member_nat(I2),aa_nat_set_nat(set_ord_atMost_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),K2)) ) ).

% atMost_iff
tff(fact_7704_atMost__iff,axiom,
    ! [I2: int,K2: int] :
      ( pp(aa_set_int_bool(member_int(I2),set_ord_atMost_int(K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K2)) ) ).

% atMost_iff
tff(fact_7705_finite__atMost,axiom,
    ! [K2: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K2))) ).

% finite_atMost
tff(fact_7706_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X),set_or58775011639299419et_int(Y))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% atMost_subset_iff
tff(fact_7707_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(X)),set_ord_atMost_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_7708_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(X)),set_ord_atMost_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% atMost_subset_iff
tff(fact_7709_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_7710_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X)),set_ord_atMost_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% atMost_subset_iff
tff(fact_7711_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_7712_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = zero_zero_complex )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_7713_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_7714_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_7715_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_7716_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_7717_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_7718_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_7719_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,H),set_or58775011639299419et_int(H2))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(L),H))
        | pp(aa_set_int_bool(ord_less_eq_set_int(H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7720_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H)),set_ord_atMost_rat(H2)))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7721_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(L,H)),set_ord_atMost_num(H2)))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7722_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(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)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7723_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H)),set_ord_atMost_int(H2)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7724_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(H2)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7725_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_7726_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_7727_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_7728_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_7729_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_7730_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_7731_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_7732_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_7733_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_7734_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_7735_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_7736_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_7737_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_7738_infinite__Iic,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_atMost_int(A))) ).

% infinite_Iic
tff(fact_7739_not__Iic__eq__Icc,axiom,
    ! [H2: int,L: int,H: int] : set_ord_atMost_int(H2) != aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),H) ).

% not_Iic_eq_Icc
tff(fact_7740_not__Iic__eq__Icc,axiom,
    ! [H2: real,L: real,H: real] : set_ord_atMost_real(H2) != set_or1222579329274155063t_real(L,H) ).

% not_Iic_eq_Icc
tff(fact_7741_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ug,U)) ).

% atMost_def
tff(fact_7742_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_av,U)) ).

% atMost_def
tff(fact_7743_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_ay(U)) ).

% atMost_def
tff(fact_7744_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_uh,U)) ).

% atMost_def
tff(fact_7745_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_ui,U)) ).

% atMost_def
tff(fact_7746_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bc,U)) ).

% atMost_def
tff(fact_7747_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_uj,U)) ).

% atMost_def
tff(fact_7748_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N) ).

% atMost_atLeast0
tff(fact_7749_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_ord_atMost_nat,K2) ).

% lessThan_Suc_atMost
tff(fact_7750_atMost__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2)) = insert_nat(aa_nat_nat(suc,K2),aa_nat_set_nat(set_ord_atMost_nat,K2)) ).

% atMost_Suc
tff(fact_7751_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(H)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L2),H2))) ).

% not_Iic_le_Icc
tff(fact_7752_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(H)),set_or1222579329274155063t_real(L2,H2))) ).

% not_Iic_le_Icc
tff(fact_7753_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_atMost_nat,K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_7754_frac__ge__0,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_ge_0
tff(fact_7755_frac__ge__0,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))) ).

% frac_ge_0
tff(fact_7756_frac__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real)) ).

% frac_lt_1
tff(fact_7757_frac__lt__1,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat)) ).

% frac_lt_1
tff(fact_7758_frac__1__eq,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X) ).

% frac_1_eq
tff(fact_7759_frac__1__eq,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).

% frac_1_eq
tff(fact_7760_atMost__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(numeral_numeral_nat,K2),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K2))) ).

% atMost_nat_numeral
tff(fact_7761_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(A)),set_ord_lessThan_rat(B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_7762_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(A)),set_ord_lessThan_num(B)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_7763_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_7764_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_7765_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(A)),set_or5984915006950818249n_real(B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_7766_norm__less__p1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,X))),one_one_real)))) ).

% norm_less_p1
tff(fact_7767_norm__less__p1,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X))),one_one_complex)))) ).

% norm_less_p1
tff(fact_7768_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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_la(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7769_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7770_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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_kq(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7771_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7772_sum__telescope,axiom,
    ! [F: fun_nat_int,I2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_nk(F)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = 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,I2))) ).

% sum_telescope
tff(fact_7773_sum__telescope,axiom,
    ! [F: fun_nat_rat,I2: nat] : groups2906978787729119204at_rat(aTP_Lamm_nl(F),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_7774_sum__telescope,axiom,
    ! [F: fun_nat_real,I2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nm(F)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_7775_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_complex,N: nat,D: fun_nat_complex] :
      ( ! [X5: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),X5),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(D),X5),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_complex(C2,I4) = aa_nat_complex(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_7776_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_real,N: nat,D: fun_nat_real] :
      ( ! [X5: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),X5)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(D),X5)),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_real(C2,I4) = aa_nat_real(D,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_7777_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_rv(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7778_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7779_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7780_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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_la(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7781_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = 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_kq(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7782_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_um(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_7783_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_up(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ur(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_7784_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_us(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ut(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_7785_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_uu(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uv(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_7786_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uw,R2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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),N))),N) ).

% sum_choose_lower
tff(fact_7787_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ux,N)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = 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,N),M2)),one_one_nat)),M2) ).

% choose_rising_sum(2)
tff(fact_7788_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ux,N)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = 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,N),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_7789_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% frac_eq
tff(fact_7790_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).

% frac_eq
tff(fact_7791_polyfun__eq__0,axiom,
    ! [C2: fun_nat_complex,N: nat] :
      ( ! [X5: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),X5),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_complex(C2,I4) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_7792_polyfun__eq__0,axiom,
    ! [C2: fun_nat_real,N: nat] :
      ( ! [X5: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),X5)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_real(C2,I4) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_7793_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_complex,N: nat,K2: nat] :
      ( ! [W: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),W),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( aa_nat_complex(C2,K2) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_7794_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_real,N: nat,K2: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),W)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( aa_nat_real(C2,K2) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_7795_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real) ) ) ) ).

% frac_add
tff(fact_7796_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat) ) ) ) ).

% frac_add
tff(fact_7797_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_la(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7798_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7799_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_kq(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7800_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7801_sum__up__index__split,axiom,
    ! [F: fun_nat_int,M2: nat,N: 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,M2),N))) = 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,M2))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_7802_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,M2: nat,N: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,M2))),groups2906978787729119204at_rat(F,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_7803_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,M2: nat,N: 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,M2),N))) = 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,M2))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_7804_sum__up__index__split,axiom,
    ! [F: fun_nat_real,M2: nat,N: 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,M2),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_7805_prod_OatMost__shift,axiom,
    ! [G: fun_nat_complex,N: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,zero_zero_nat)),groups6464643781859351333omplex(aTP_Lamm_rv(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_7806_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_kr(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_7807_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_lb(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_7808_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : aa_set_nat_int(groups705719431365010083at_int(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_la(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_7809_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_kq(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_7810_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_7811_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uy(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_va(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.triangle_reindex_eq
tff(fact_7812_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uy(N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vc(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.triangle_reindex_eq
tff(fact_7813_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uy(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ve(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_7814_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uy(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_vf(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_7815_sum__choose__diagonal,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vg(M2),N)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M2) ) ) ).

% sum_choose_diagonal
tff(fact_7816_vandermonde,axiom,
    ! [M2: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vh(M2),N),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,M2),N)),R2) ).

% vandermonde
tff(fact_7817_sum__gp__basic,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_7818_sum__gp__basic,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_7819_sum__gp__basic,axiom,
    ! [X: int,N: 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),X)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_7820_sum__gp__basic,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_7821_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_vi(C2),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_complex(C2,I4) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_7822_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_vj(C2),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_real(C2,I4) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_7823_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_complex,K2: nat,N: nat] :
      ( ( aa_nat_complex(C2,K2) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_vi(C2),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_7824_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_real,K2: nat,N: nat] :
      ( ( aa_nat_real(C2,K2) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_vj(C2),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_7825_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_complex,A: complex,N: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_7826_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_rat,A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_7827_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_int,A: int,N: nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vl(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vl(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_vl(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_7828_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_real,A: real,N: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_7829_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_complex,N: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_uk(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_7830_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_rat,N: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(C2),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_7831_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_int,N: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vl(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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_vl(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vl(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_7832_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_real,N: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_7833_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M2)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_7834_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M2)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_7835_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M2)),aa_set_nat_int(groups3539618377306564664at_int(power_power_int(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_7836_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_7837_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_vm(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_va(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.triangle_reindex
tff(fact_7838_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_vm(N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vc(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.triangle_reindex
tff(fact_7839_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_vm(N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ve(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_7840_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_vm(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_vf(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_7841_binomial,axiom,
    ! [A: nat,B: nat,N: 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)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vn(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_7842_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: 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,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_lw(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_7843_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_lx(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_7844_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ly(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_7845_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: 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,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lz(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_7846_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_complex,N: nat,B: fun_nat_complex,X: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_complex(A,I3) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(A),X),aa_nat_set_nat(set_ord_atMost_nat,M2))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(B),X),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_vp(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_7847_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_rat,N: nat,B: fun_nat_rat,X: rat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_rat(A,I3) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(A),X),aa_nat_set_nat(set_ord_atMost_nat,M2))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(B),X),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_vr(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_7848_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_int,N: nat,B: fun_nat_int,X: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_int(A,I3) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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_vl(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vl(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_vt(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_7849_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_real,N: nat,B: fun_nat_real,X: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_real(A,I3) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_vv(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_7850_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_complex,N: 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,aa_num_num(bit0,one))),N)))) = groups6464643781859351333omplex(aTP_Lamm_sj(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_7851_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: 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,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_sk(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_7852_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_sl(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_7853_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: 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,aa_num_num(bit0,one))),N)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_sm(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_7854_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_sn(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_7855_polyfun__eq__const,axiom,
    ! [C2: fun_nat_complex,N: nat,K2: complex] :
      ( ! [X5: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(C2),X5),aa_nat_set_nat(set_ord_atMost_nat,N)) = K2
    <=> ( ( aa_nat_complex(C2,zero_zero_nat) = K2 )
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N)))
           => ( aa_nat_complex(C2,X5) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_7856_polyfun__eq__const,axiom,
    ! [C2: fun_nat_real,N: nat,K2: real] :
      ( ! [X5: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(C2),X5)),aa_nat_set_nat(set_ord_atMost_nat,N)) = K2
    <=> ( ( aa_nat_real(C2,zero_zero_nat) = K2 )
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N)))
           => ( aa_nat_real(C2,X5) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_7857_binomial__ring,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),N) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vw(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_7858_binomial__ring,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A),B)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_vx(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_7859_binomial__ring,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vy(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_7860_binomial__ring,axiom,
    ! [A: nat,B: nat,N: 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)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vn(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_7861_binomial__ring,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_vz(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_7862_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),B)),N) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wa(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_7863_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),B)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_wb(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_7864_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wc(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_7865_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wd(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_7866_polynomial__product__nat,axiom,
    ! [M2: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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_we(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_we(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wg(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_7867_floor__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_7868_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_7869_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wh(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wi(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7870_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wj(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wk(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7871_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wl(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wm(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7872_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wn(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wo(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7873_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wp(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wq(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7874_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wr(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wq(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7875_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ws(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wi(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7876_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wt(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wk(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7877_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wu(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wm(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7878_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wv(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wo(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7879_root__polyfun,axiom,
    ! [N: nat,Z: complex,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_complex(power_power_complex(Z),N) = A )
      <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ww(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_7880_root__polyfun,axiom,
    ! [N: nat,Z: int,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_int(power_power_int(Z),N) = A )
      <=> ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wx(N),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_7881_root__polyfun,axiom,
    ! [N: nat,Z: rat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_rat(power_power_rat(Z),N) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_wy(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_7882_root__polyfun,axiom,
    ! [N: nat,Z: real,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wz(N),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_7883_sum__gp0,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp0
tff(fact_7884_sum__gp0,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp0
tff(fact_7885_sum__gp0,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp0
tff(fact_7886_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_xa(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_7887_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_xb(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_7888_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xc,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_7889_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xd,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_7890_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_xf(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7891_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_rat,X: rat,Y: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_xh(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7892_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vl(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vl(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_xj(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7893_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_xl(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7894_binomial__r__part__sum,axiom,
    ! [M2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ).

% binomial_r_part_sum
tff(fact_7895_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups2073611262835488442omplex(aTP_Lamm_xm(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_7896_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_xn(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_7897_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xo,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_7898_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xp,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_7899_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_7900_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_7901_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_7902_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel_iff
tff(fact_7903_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))) ) ).

% exp_less_mono
tff(fact_7904_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_less_exp_iff
tff(fact_7905_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_7906_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).

% exp_ln
tff(fact_7907_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% exp_ln_iff
tff(fact_7908_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel
tff(fact_7909_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X4: real] : aa_real_real(exp_real,X4) = Y ) ).

% exp_total
tff(fact_7910_exp__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_gt_zero
tff(fact_7911_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_less_zero
tff(fact_7912_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X))) ) ).

% exp_gt_one
tff(fact_7913_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_7914_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7915_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7916_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X))) ) ) ).

% cot_gt_zero
tff(fact_7917_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_7918_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_7919_binomial__code,axiom,
    ! [N: nat,K2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K2)) ) ) ) ) ) ).

% binomial_code
tff(fact_7920_modulo__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,M2),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_7921_powr__int,axiom,
    ! [X: real,I2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,I2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I2)))) ) ) ) ) ).

% powr_int
tff(fact_7922_nat__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).

% nat_int
tff(fact_7923_nat__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_numeral
tff(fact_7924_nat__of__bool,axiom,
    ! [P2: bool] : aa_int_nat(nat2,aa_bool_int(zero_n2684676970156552555ol_int,P2)) = aa_bool_nat(zero_n2687167440665602831ol_nat,P2) ).

% nat_of_bool
tff(fact_7925_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_7926_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_7927_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa_int_nat(nat2,I2) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_7928_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% zless_nat_conj
tff(fact_7929_nat__neg__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_7930_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_7931_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_7932_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_7933_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_7934_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_7935_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_int_int(sgn_sgn_int,R2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_7936_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% zero_less_nat_eq
tff(fact_7937_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_7938_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_7939_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_7940_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_7941_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_7942_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_7943_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_7944_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_7945_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_7946_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_7947_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_7948_fact__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_7949_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_7950_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_7951_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_7952_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N3: nat,L3: int] : K2 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N3)) ).

% int_sgnE
tff(fact_7953_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z4))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_7954_all__nat,axiom,
    ! [P2: fun_nat_bool] :
      ( ! [X_13: nat] : pp(aa_nat_bool(P2,X_13))
    <=> ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => pp(aa_nat_bool(P2,aa_int_nat(nat2,X5))) ) ) ).

% all_nat
tff(fact_7955_ex__nat,axiom,
    ! [P2: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P2,X_13))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
          & pp(aa_nat_bool(P2,aa_int_nat(nat2,X5))) ) ) ).

% ex_nat
tff(fact_7956_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_7957_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_7958_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_mono_iff
tff(fact_7959_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_7960_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_7961_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_7962_int__eq__iff,axiom,
    ! [M2: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = Z )
    <=> ( ( M2 = aa_int_nat(nat2,Z) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).

% int_eq_iff
tff(fact_7963_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_7964_nat__plus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_7965_fact__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ) ) ).

% fact_diff_Suc
tff(fact_7966_zsgn__def,axiom,
    ! [I2: int] :
      ( ( ( I2 = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I2) = zero_zero_int ) )
      & ( ( I2 != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I2))
           => ( aa_int_int(sgn_sgn_int,I2) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I2))
           => ( aa_int_int(sgn_sgn_int,I2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_7967_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_7968_binomial__fact__lemma,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_7969_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_7970_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_7971_nat__eq__iff,axiom,
    ! [W2: int,M2: nat] :
      ( ( aa_int_nat(nat2,W2) = M2 )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M2) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_7972_nat__eq__iff2,axiom,
    ! [M2: nat,W2: int] :
      ( ( M2 = aa_int_nat(nat2,W2) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M2) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_7973_split__nat,axiom,
    ! [P2: fun_nat_bool,I2: int] :
      ( pp(aa_nat_bool(P2,aa_int_nat(nat2,I2)))
    <=> ( ! [N2: nat] :
            ( ( I2 = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => pp(aa_nat_bool(P2,N2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),zero_zero_int))
         => pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_7974_le__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K2)) ) ) ).

% le_nat_iff
tff(fact_7975_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z4))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_7976_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_7977_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z4),Z))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_7978_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_7979_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_7980_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_7981_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_7982_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_7983_fact__eq__fact__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ce),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,N)),M2))) ) ) ).

% fact_eq_fact_times
tff(fact_7984_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_7985_binomial__altdef__nat,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_7986_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_7987_nat__less__iff,axiom,
    ! [W2: int,M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),M2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M2))) ) ) ).

% nat_less_iff
tff(fact_7988_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_7989_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_7990_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ce),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M2)) ) ) ).

% fact_div_fact
tff(fact_7991_diff__nat__eq__if,axiom,
    ! [Z4: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4))) ) ) ) ).

% diff_nat_eq_if
tff(fact_7992_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real))
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ul(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7993_nat__dvd__iff,axiom,
    ! [Z: int,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M2))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_7994_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_xq(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_7995_divide__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,M2),N)),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_7996_sin__coeff__def,axiom,
    ! [X3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3))
       => ( aa_nat_real(sin_coeff,X3) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3))
       => ( aa_nat_real(sin_coeff,X3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3)) ) ) ) ).

% sin_coeff_def
tff(fact_7997_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_xr,X)) ) ).

% summable_arctan_series
tff(fact_7998_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xs,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7999_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_xt,X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_8000_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8001_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_8002_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),pi))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8003_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_8004_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero
tff(fact_8005_abs__real__def,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_real_def
tff(fact_8006_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N3))),aa_nat_real(G,N3))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_xu(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8007_sgn__real__def,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_8008_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8009_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_8010_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8011_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero2
tff(fact_8012_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_8013_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_8014_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_8015_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_8016_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8017_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8018_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8019_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xv,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8020_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xv,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8021_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval
tff(fact_8022_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_xw,X)) ) ).

% monoseq_arctan_series
tff(fact_8023_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arctan,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xr,X)) ) ) ).

% arctan_series
tff(fact_8024_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_8025_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_arctan_iff
tff(fact_8026_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% arctan_less_zero_iff
tff(fact_8027_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_8028_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_8029_dvd__nat__abs__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K2)) ) ).

% dvd_nat_abs_iff
tff(fact_8030_nat__abs__dvd__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_abs_dvd_iff
tff(fact_8031_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_8032_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% arctan_less_iff
tff(fact_8033_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone
tff(fact_8034_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M2),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M2) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_8035_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M3: int] :
        ? [N2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M3),aa_int_int(abs_abs_int,N2)))
          & pp(aa_set_int_bool(member_int(N2),S2)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_8036_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M3: int] :
        ? [N2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M3),aa_int_int(abs_abs_int,N2)))
          & pp(aa_set_int_bool(member_int(N2),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8037_zabs__def,axiom,
    ! [I2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I2) = aa_int_int(uminus_uminus_int,I2) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I2) = I2 ) ) ) ).

% zabs_def
tff(fact_8038_dvd__imp__le__int,axiom,
    ! [I2: int,D: int] :
      ( ( I2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I2))) ) ) ).

% dvd_imp_le_int
tff(fact_8039_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_8040_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K2),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_8041_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_8042_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M2),N)),M2))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_8043_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).

% nat_abs_int_diff
tff(fact_8044_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),I3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M2)),K2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_nat_int(F,N)))
           => ? [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),I3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
                & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_8045_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_8046_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_8047_arctan__ubound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% arctan_ubound
tff(fact_8048_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_nat_int(F,N)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_8049_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% arctan_bounded
tff(fact_8050_arctan__lbound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))) ).

% arctan_lbound
tff(fact_8051_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_nat_int(F,N)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_8052_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_8053_divide__int__def,axiom,
    ! [L: int,K2: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(divide_divide_int(K2),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K2),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K2),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_8054_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_8055_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T5) )
               => ( Y != aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8056_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8057_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8058_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8059_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_8060_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8061_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_8062_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8063_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_gt_zero
tff(fact_8064_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X4))) ) ) ).

% lemma_tan_total
tff(fact_8065_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = Y )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X4 ) ) ) ).

% tan_total
tff(fact_8066_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ).

% tan_monotone
tff(fact_8067_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8068_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8069_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_8070_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_8071_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero
tff(fact_8072_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_8073_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_pos_pi2_le
tff(fact_8074_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_8075_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8076_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_8077_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_8078_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_8079_arctan,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_8080_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_8081_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( ( aa_real_real(tan_real,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_8082_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
          & ( aa_real_real(tan_real,Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_8083_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xx,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8084_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xx,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8085_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T5)),aa_real_real(sin_real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8086_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = product_Pair_int_int(Q3,R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22)))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8087_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8088_forall__pos__mono__1,axiom,
    ! [P2: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P2,D4))
           => pp(aa_real_bool(P2,E)) ) )
     => ( ! [N3: nat] : pp(aa_real_bool(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P2,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_8089_forall__pos__mono,axiom,
    ! [P2: fun_real_bool,E2: real] :
      ( ! [D4: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D4),E))
         => ( pp(aa_real_bool(P2,D4))
           => pp(aa_real_bool(P2,E)) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => pp(aa_real_bool(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P2,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_8090_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N2: nat] :
          ( ( N2 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),E2)) ) ) ).

% real_arch_inverse
tff(fact_8091_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).

% ln_inverse
tff(fact_8092_log__inverse,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) ) ) ) ) ).

% log_inverse
tff(fact_8093_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X)))) ) ).

% plus_inverse_ge_2
tff(fact_8094_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8095_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q2: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(Q2,R2)))
    <=> ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_8096_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L)))
       => ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R2) )
         => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(Q2,R2))) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8097_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,N)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_8098_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q5,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4)))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8099_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_xy,Y)) ).

% arctan_def
tff(fact_8100_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8101_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8102_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% sinh_real_less_iff
tff(fact_8103_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% sinh_real_neg_iff
tff(fact_8104_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% sinh_real_pos_iff
tff(fact_8105_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_less_cosh_real
tff(fact_8106_cosh__real__pos,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_pos
tff(fact_8107_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_8108_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_8109_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_8110_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8111_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_8112_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_8113_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8114_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),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,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_8115_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_8116_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_bool(Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_8117_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_8118_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_8119_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_8120_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_8121_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aa_real_fun_int_bool(aTP_Lamm_xz,X)) ).

% floor_real_def
tff(fact_8122_xor__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = N ) )
      & ( ( M2 != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = M2 ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_8123_xor__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_8124_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_ya(X)) ).

% floor_rat_def
tff(fact_8125_or__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = N ) )
      & ( ( M2 != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = M2 ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_8126_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K2),L)),zero_zero_int))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% xor_negative_int_iff
tff(fact_8127_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_8128_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_8129_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_8130_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_8131_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_8132_abs__rat__def,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_rat_def
tff(fact_8133_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
     => ~ ! [S3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S3))
           => ! [T5: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T5))
               => ( R2 != aa_rat_rat(plus_plus_rat(S3),T5) ) ) ) ) ).

% obtain_pos_sum
tff(fact_8134_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_8135_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_8136_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_8137_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_8138_or__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_8139_normalize__negative,axiom,
    ! [Q2: int,P: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q2),zero_zero_int))
     => ( normalize(product_Pair_int_int(P,Q2)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_8140_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),N)) = 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,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_8141_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_8142_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M3))
         => ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),N2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X6,M3)),aa_nat_real(X6,N2)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_8143_sum__power2,axiom,
    ! [K2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),one_one_nat) ).

% sum_power2
tff(fact_8144_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K2),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% or_negative_int_iff
tff(fact_8145_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U))) ).

% finite_atLeastLessThan
tff(fact_8146_atLeastLessThan__singleton,axiom,
    ! [M2: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),aa_nat_nat(suc,M2)) = insert_nat(M2,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_8147_all__nat__less__eq,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
         => pp(aa_nat_bool(P2,M3)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
         => pp(aa_nat_bool(P2,X5)) ) ) ).

% all_nat_less_eq
tff(fact_8148_ex__nat__less__eq,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
          & pp(aa_nat_bool(P2,M3)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
          & pp(aa_nat_bool(P2,X5)) ) ) ).

% ex_nat_less_eq
tff(fact_8149_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_8150_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N) ).

% lessThan_atLeast0
tff(fact_8151_atLeastLessThan0,axiom,
    ! [M2: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_8152_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_8153_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8154_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_8155_prod__Suc__Suc__fact,axiom,
    ! [N: 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)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_8156_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_8157_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P: int,Q2: int] :
      ( ( normalize(R2) = product_Pair_int_int(P,Q2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).

% normalize_denom_pos
tff(fact_8158_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K2: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_num_nat(pred_numeral,K2)))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),aa_num_nat(pred_numeral,K2))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_num_nat(pred_numeral,K2)))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),aa_num_nat(numeral_numeral_nat,K2)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_8159_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_8160_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8161_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2))) ) )
     => ( ! [I3: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I3))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_yb(A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))),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),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8162_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_8163_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U))) ).

% finite_atLeastLessThan_int
tff(fact_8164_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_8165_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_8166_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_8167_card__lessThan,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_8168_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,N))) = N ).

% card_Collect_less_nat
tff(fact_8169_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_8170_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_8171_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bc,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_8172_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_8173_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K2)),zero_zero_int))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_8174_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_8175_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U)) = 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,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_8176_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_8177_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_8178_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_8179_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).

% not_bit_Suc_0_numeral
tff(fact_8180_card__less__Suc2,axiom,
    ! [M5: set_nat,I2: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_yc(M5),I2))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_yd(M5),I2))) ) ) ).

% card_less_Suc2
tff(fact_8181_card__less__Suc,axiom,
    ! [M5: set_nat,I2: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_yc(M5),I2)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_yd(M5),I2))) ) ) ).

% card_less_Suc
tff(fact_8182_card__less,axiom,
    ! [M5: set_nat,I2: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_yd(M5),I2))) != zero_zero_nat ) ) ).

% card_less
tff(fact_8183_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4)))))
     => ( A4 = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_8184_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M2: nat,K2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),K2))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_8185_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N4)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_8186_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce),S2))) ).

% card_sum_le_nat_sum
tff(fact_8187_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_ye(C2),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_8188_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_dc,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_8189_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N3: nat] :
          ( ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),M4))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),M4))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N3)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N3),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N3)) ) ) ) ).

% int_bit_bound
tff(fact_8190_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_bk),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_8191_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_yf),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_8192_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M2)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_8193_max__Suc2,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_yg,N)),M2) ).

% max_Suc2
tff(fact_8194_max__Suc1,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M2) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_yh,N)),M2) ).

% max_Suc1
tff(fact_8195_diff__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ce),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ).

% diff_Suc
tff(fact_8196_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ce),Nat) ).

% pred_def
tff(fact_8197_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_weak))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_weak)) ) ) ) ).

% wmin_insertI
tff(fact_8198_push__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N),K2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_8199_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_8200_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_weak)) ).

% wmin_emptyI
tff(fact_8201_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K2: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M2),K2)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% bit_push_bit_iff_int
tff(fact_8202_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M2),Q2)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8203_binomial__def,axiom,
    ! [N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_yi(N),K2))) ).

% binomial_def
tff(fact_8204_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),YS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_max_weak))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS)),fun_max_weak)) ) ) ) ).

% wmax_insertI
tff(fact_8205_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),Y6))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,Y6)),fun_max_strict))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X6),Y6)),fun_max_strict)) ) ) ) ).

% smax_insertI
tff(fact_8206_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_strict))
         => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_strict)) ) ) ) ).

% smin_insertI
tff(fact_8207_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X6))
     => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X6)),fun_max_weak)) ) ).

% wmax_emptyI
tff(fact_8208_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_strict)) ) ).

% smin_emptyI
tff(fact_8209_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,Y6))
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y6)),fun_max_strict)) ) ) ).

% smax_emptyI
tff(fact_8210_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_8211_prod__decode__aux_Osimps,axiom,
    ! [M2: nat,K2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),K2))
       => ( nat_prod_decode_aux(K2,M2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),M2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),K2))
       => ( nat_prod_decode_aux(K2,M2) = nat_prod_decode_aux(aa_nat_nat(suc,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,K2))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_8212_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_8213_Suc__0__div__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_div_numeral
tff(fact_8214_Suc__0__mod__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_mod_numeral
tff(fact_8215_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [R3: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S2))))
          & ! [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S2)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N5)),S2)) ) ) ) ).

% finite_enumerate
tff(fact_8216_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_8217_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_8218_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(one_one_int,zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_8219_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% bezw.pelims
tff(fact_8220_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_8221_normalize__def,axiom,
    ! [P: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P)))
       => ( normalize(P) = product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P)),product_snd_int_int(P))),aa_int_int(divide_divide_int(product_snd_int_int(P)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P)),product_snd_int_int(P)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P)))
       => ( ( ( product_snd_int_int(P) = zero_zero_int )
           => ( normalize(P) = product_Pair_int_int(zero_zero_int,one_one_int) ) )
          & ( ( product_snd_int_int(P) != zero_zero_int )
           => ( normalize(P) = product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P)),product_snd_int_int(P)))),aa_int_int(divide_divide_int(product_snd_int_int(P)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P)),product_snd_int_int(P))))) ) ) ) ) ) ).

% normalize_def
tff(fact_8222_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N),K2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_8223_gcd__pos__int,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M2),N)))
    <=> ( ( M2 != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_8224_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_8225_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A)) ) ).

% gcd_le1_int
tff(fact_8226_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B)) ) ).

% gcd_le2_int
tff(fact_8227_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(modulo_modulo_int(X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_8228_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_8229_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_8230_gcd__0__left__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

% gcd_0_left_nat
tff(fact_8231_gcd__0__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

% gcd_0_nat
tff(fact_8232_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_8233_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_8234_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_8235_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_8236_gcd__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_8237_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N)))
    <=> ( ( M2 != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_8238_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U))) ).

% finite_greaterThanLessThan_int
tff(fact_8239_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_8240_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_8241_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_8242_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_8243_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_8244_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_8245_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_8246_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_8247_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_8248_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_8249_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_8250_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_8251_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_8252_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_8253_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_8254_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_8255_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_8256_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ) ).

% gcd_le2_nat
tff(fact_8257_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ) ).

% gcd_le1_nat
tff(fact_8258_gcd__diff1__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N) ) ) ).

% gcd_diff1_nat
tff(fact_8259_gcd__diff2__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N) ) ) ).

% gcd_diff2_nat
tff(fact_8260_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_8261_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_8262_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_8263_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_8264_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_8265_real__root__power,axiom,
    ! [N: nat,X: real,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),K2) ) ) ).

% real_root_power
tff(fact_8266_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).

% real_root_abs
tff(fact_8267_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_8268_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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) ) )
      | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_8269_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_8270_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_8271_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_gt_zero
tff(fact_8272_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_8273_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_8274_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_pos_pos
tff(fact_8275_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_8276_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_8277_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_8278_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_8279_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_8280_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_8281_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_8282_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_8283_ln__root,axiom,
    ! [N: nat,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_8284_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_8285_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,N),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% log_base_root
tff(fact_8286_split__root,axiom,
    ! [P2: fun_real_bool,N: nat,X: real] :
      ( pp(aa_real_bool(P2,aa_real_real(aa_nat_fun_real_real(root,N),X)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P2,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),N)) = X )
             => pp(aa_real_bool(P2,Y5)) ) ) ) ) ).

% split_root
tff(fact_8287_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% gcd_nat.pelims
tff(fact_8288_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bc,aTP_Lamm_bb) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_8289_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,L),U))) ).

% finite_greaterThanLessThan
tff(fact_8290_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_8291_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_8292_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_8293_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_yj) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_8294_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_8295_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5)) ) ).

% Inf_nat_def1
tff(fact_8296_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_yl),Xa2),X)) ).

% times_int.abs_eq
tff(fact_8297_int_Oabs__induct,axiom,
    ! [P2: fun_int_bool,X: int] :
      ( ! [Y3: product_prod_nat_nat] : pp(aa_int_bool(P2,aa_Pro4240683299535347697at_int(abs_Integ,Y3)))
     => pp(aa_int_bool(P2,X)) ) ).

% int.abs_induct
tff(fact_8298_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)) ).

% eq_Abs_Integ
tff(fact_8299_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).

% nat.abs_eq
tff(fact_8300_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_8301_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_8302_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ym),X)) ).

% uminus_int.abs_eq
tff(fact_8303_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_8304_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_yo),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_8305_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_yq),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_8306_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ys),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_8307_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_yu),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_8308_max__rpair__set,axiom,
    pp(aa_Pro4324259192382070444t_bool(fun_re2478310338295953701at_nat,produc9060074326276436823at_nat(fun_max_strict,fun_max_weak))) ).

% max_rpair_set
tff(fact_8309_min__rpair__set,axiom,
    pp(aa_Pro4324259192382070444t_bool(fun_re2478310338295953701at_nat,produc9060074326276436823at_nat(fun_min_strict,fun_min_weak))) ).

% min_rpair_set
tff(fact_8310_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_yq),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_8311_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_yo),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_8312_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_yv) ).

% prod_encode_def
tff(fact_8313_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M5,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_8314_prod__encode__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(nat_prod_encode,X) = aa_Pro4243173770044397973at_nat(nat_prod_encode,Y) )
    <=> ( X = Y ) ) ).

% prod_encode_eq
tff(fact_8315_Gcd__in,axiom,
    ! [A4: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_set_nat_bool(member_nat(A3),A4))
         => ( pp(aa_set_nat_bool(member_nat(B3),A4))
           => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A4)) ) )
     => ( ( A4 != bot_bot_set_nat )
       => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A4)),A4)) ) ) ).

% Gcd_in
tff(fact_8316_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_2
tff(fact_8317_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_1
tff(fact_8318_nat_Orep__eq,axiom,
    ! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).

% nat.rep_eq
tff(fact_8319_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M2: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K2,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),M2) ).

% prod_encode_prod_decode_aux
tff(fact_8320_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_ym)) ).

% uminus_int_def
tff(fact_8321_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_yl)) ).

% times_int_def
tff(fact_8322_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_yu)) ).

% minus_int_def
tff(fact_8323_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ys)) ).

% plus_int_def
tff(fact_8324_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yw,C2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,X),Y)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yw,C2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,X),Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yw,C2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_8325_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(A),B))),aa_int_rat(fract(A),B)))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int)))) ) ).

% rat_floor_lemma
tff(fact_8326_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I2),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,I2)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_8327_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,I2)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_8328_less__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

% less_rat
tff(fact_8329_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A4))) ).

% zero_notin_Suc_image
tff(fact_8330_Rat__induct__pos,axiom,
    ! [P2: fun_rat_bool,Q2: rat] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => pp(aa_rat_bool(P2,aa_int_rat(fract(A3),B3))) )
     => pp(aa_rat_bool(P2,Q2)) ) ).

% Rat_induct_pos
tff(fact_8331_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_8332_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_8333_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_8334_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_8335_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_8336_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_8337_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% Fract_less_zero_iff
tff(fact_8338_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% zero_less_Fract_iff
tff(fact_8339_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% one_less_Fract_iff
tff(fact_8340_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% Fract_less_one_iff
tff(fact_8341_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_Fract_iff
tff(fact_8342_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(fract(A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% Fract_le_zero_iff
tff(fact_8343_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(fract(A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% Fract_le_one_iff
tff(fact_8344_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% one_le_Fract_iff
tff(fact_8345_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I2))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,I2),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_8346_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_8347_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_yo)) ).

% less_int_def
tff(fact_8348_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_yq)) ).

% less_eq_int_def
tff(fact_8349_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_atMost_int(K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_8350_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_lessThan_int(K3))) ) ).

% finite_int_iff_bounded
tff(fact_8351_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_8352_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastAtMost
tff(fact_8353_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastLessThan
tff(fact_8354_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_yx(L),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U) ).

% image_add_int_atLeastLessThan
tff(fact_8355_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_8356_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_8357_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = insert_real(zero_zero_real,bot_bot_set_real) ) )
      & ( ( A != zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_8358_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_8359_infinite__UNIV__nat,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% infinite_UNIV_nat
tff(fact_8360_nat__not__finite,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% nat_not_finite
tff(fact_8361_infinite__UNIV__int,axiom,
    ~ pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ).

% infinite_UNIV_int
tff(fact_8362_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_8363_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_8364_int__in__range__abs,axiom,
    ! [N: nat] : pp(aa_set_int_bool(member_int(aa_nat_int(semiri1314217659103216013at_int,N)),image_int_int(abs_abs_int,top_top_set_int))) ).

% int_in_range_abs
tff(fact_8365_UNIV__nat__eq,axiom,
    top_top_set_nat = insert_nat(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_8366_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yy,N),top_top_set_nat) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_8367_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_yz,N)),X) ) ) ) ).

% root_def
tff(fact_8368_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_8369_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_8370_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_8371_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_8372_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_8373_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_set_real_bool(member_real(X),set_or1633881224788618240n_real(A,B)))
       => ( pp(aa_set_real_bool(member_real(Y),set_or1633881224788618240n_real(A,B)))
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(A,B)))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_8374_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_8375_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_8376_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_8377_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_8378_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_8379_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_8380_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F5,Z2)) ) ) ) ) ).

% MVT2
tff(fact_8381_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => ( aa_real_real(F,X) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_8382_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_8383_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,X))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_8384_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_8385_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_8386_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_za,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_8387_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_zb,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_8388_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_8389_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M2: real,X: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M2,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_zc(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M2),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_8390_DERIV__powr,axiom,
    ! [G: fun_real_real,M2: real,X: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M2,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_zd(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(F,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M2),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_8391_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A4: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_8392_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_8393_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D5: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D5 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D5 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_8394_artanh__real__has__field__derivative,axiom,
    ! [X: real,A4: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_8395_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_ze(F),X4)) )
     => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
         => has_fi5821293074295781190e_real(aTP_Lamm_zg(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ze(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_8396_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_8397_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_8398_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: 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] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_zh(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_8399_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: 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] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_zh(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_8400_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_8401_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T5: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_zi(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_8402_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_zi(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_8403_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T5: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_zi(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_8404_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X != 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] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_zh(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_8405_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,T5: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_zh(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_8406_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
             => ? [T5: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),C2))
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_zj(Diff),A),C2)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_8407_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
             => ? [T5: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),T5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),B))
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_zj(Diff),B),C2)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_8408_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
                 => ( ( X != C2 )
                   => ? [T5: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C2))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T5))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),C2)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C2))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),T5))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_zk(Diff),C2),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_8409_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K2: nat,B4: real] :
      ( ! [M: nat,T5: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)) )
     => ( ( N = aa_nat_nat(suc,K2) )
       => ! [M4: nat,T6: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_zm(N),Diff),B4),M4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T6)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_zn(Diff),M4),T6)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M4))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M4)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M4))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_8410_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_zo,suminf_real(aa_real_fun_nat_real(aTP_Lamm_zp,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_8411_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D5: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X != zero_zero_real )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D5 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
               => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_8412_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_8413_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_8414_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_8415_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_8416_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_8417_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_zq,M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_8418_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_8419_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X3))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_8420_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X3))),R3)) )
               => ( aa_real_real(F,X3) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_8421_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X3))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X3))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_8422_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2)))) ) ).

% card_le_Suc_Max
tff(fact_8423_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      ( ( ( X6 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = zero_zero_nat ) )
      & ( ( X6 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X6) ) ) ) ).

% Sup_nat_def
tff(fact_8424_divide__nat__def,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zr(N),M2))) ) ) ) ).

% divide_nat_def
tff(fact_8425_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zs(N),M2))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_8426_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat)))
         => ! [N5: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_zt(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_8427_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real))
         => ! [N5: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_zt(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_8428_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_zt(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_8429_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_8430_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_8431_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zu,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_8432_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X6,N3))) )
     => filterlim_nat_real(aTP_Lamm_zv(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_8433_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zw,C2),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_8434_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,aa_nat_nat(suc,N3))))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),L))
       => ( ! [E: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
             => ? [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N5)),E))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_8435_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_8436_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zx(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_8437_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C2)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_8438_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_8439_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zy,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_8440_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_xw,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_8441_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))),L3))
              & filterlim_nat_real(aTP_Lamm_zz(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_aaa(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_8442_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_aaa(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_zt(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_8443_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => filterlim_nat_real(aTP_Lamm_aaa(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_zt(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_8444_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X4))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_8445_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aab(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_8446_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aab(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_8447_eventually__sequentially__seg,axiom,
    ! [P2: fun_nat_bool,K2: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_aac(P2),K2),at_top_nat)
    <=> eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_8448_eventually__sequentiallyI,axiom,
    ! [C2: nat,P2: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),X4))
         => pp(aa_nat_bool(P2,X4)) )
     => eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_8449_eventually__sequentially,axiom,
    ! [P2: fun_nat_bool] :
      ( eventually_nat(P2,at_top_nat)
    <=> ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N2))
         => pp(aa_nat_bool(P2,N2)) ) ) ).

% eventually_sequentially
tff(fact_8450_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),F3) ) ).

% le_sequentially
tff(fact_8451_sequentially__offset,axiom,
    ! [P2: fun_nat_bool,K2: nat] :
      ( eventually_nat(P2,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_aac(P2),K2),at_top_nat) ) ).

% sequentially_offset
tff(fact_8452_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aad(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_8453_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_8454_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,L),U))) ).

% finite_greaterThanAtMost
tff(fact_8455_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_8456_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_8457_GreatestI__ex__nat,axiom,
    ! [P2: fun_nat_bool,B: nat] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P2,X_1))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P2,order_Greatest_nat(P2))) ) ) ).

% GreatestI_ex_nat
tff(fact_8458_Greatest__le__nat,axiom,
    ! [P2: fun_nat_bool,K2: nat,B: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),order_Greatest_nat(P2))) ) ) ).

% Greatest_le_nat
tff(fact_8459_GreatestI__nat,axiom,
    ! [P2: fun_nat_bool,K2: nat,B: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P2,order_Greatest_nat(P2))) ) ) ).

% GreatestI_nat
tff(fact_8460_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,I2),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_8461_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U))) ).

% finite_greaterThanAtMost_int
tff(fact_8462_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_8463_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_8464_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
               => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_8465_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_8466_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D5: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D5,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D5 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
           => ( ! [Y3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B))
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D5),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_8467_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_8468_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_8469_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(B,X)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ).

% LIM_less_bound
tff(fact_8470_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_8471_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X))),D))
           => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_8472_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F5,C)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_8473_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_8474_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aad(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_8475_greaterThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K2)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K2),insert_nat(aa_nat_nat(suc,K2),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_8476_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_8477_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ( ! [X4: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
         => ( ! [X4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
           => ( ! [X4: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X4,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_8478_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_8479_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K2) ).

% atLeast_Suc_greaterThan
tff(fact_8480_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_8481_atLeast__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K2),insert_nat(K2,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_8482_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ? [L3: real,Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z2,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_8483_bdd__above__nat,axiom,
    ! [X6: set_nat] :
      ( pp(aa_set_nat_bool(condit2214826472909112428ve_nat,X6))
    <=> pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ).

% bdd_above_nat
tff(fact_8484_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                & ! [X3: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z2),X3) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_8485_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Xi))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi),B))
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F5,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_8486_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_8487_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_8488_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_8489_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_8490_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_8491_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_aaf,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_8492_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_aah,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_8493_eventually__prod__sequentially,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P2,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M3))
         => ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N2))
             => pp(aa_Pro2232559814647311250t_bool(P2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N2),M3))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_8494_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_8495_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_8496_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_8497_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aai,K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_8498_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ~ pp(aa_set_nat_bool(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S2))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_8499_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_8500_surj__int__encode,axiom,
    image_int_nat(nat_int_encode,top_top_set_int) = top_top_set_nat ).

% surj_int_encode
tff(fact_8501_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_real_bool(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_yz,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_8502_int__encode__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_nat(nat_int_encode,X) = aa_int_nat(nat_int_encode,Y) )
    <=> ( X = Y ) ) ).

% int_encode_eq
tff(fact_8503_log__inj,axiom,
    ! [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => pp(aa_set_real_bool(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real))) ) ).

% log_inj
tff(fact_8504_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_8505_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(inj_on2178005380612969504at_nat(nat_prod_encode),A4)) ).

% inj_prod_encode
tff(fact_8506_inj__int__encode,axiom,
    ! [A4: set_int] : pp(aa_set_int_bool(inj_on_int_nat(nat_int_encode),A4)) ).

% inj_int_encode
tff(fact_8507_inj__Suc,axiom,
    ! [N4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N4)) ).

% inj_Suc
tff(fact_8508_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K2: nat] :
      ( ! [N3: nat] :
          ( pp(aa_set_nat_bool(member_nat(N3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yw,K2)),N4)) ) ).

% inj_on_diff_nat
tff(fact_8509_inj__on__set__encode,axiom,
    pp(aa_set_set_nat_bool(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat))) ).

% inj_on_set_encode
tff(fact_8510_surj__int__decode,axiom,
    image_nat_int(nat_int_decode,top_top_set_nat) = top_top_set_int ).

% surj_int_decode
tff(fact_8511_int__encode__inverse,axiom,
    ! [X: int] : aa_nat_int(nat_int_decode,aa_int_nat(nat_int_encode,X)) = X ).

% int_encode_inverse
tff(fact_8512_int__decode__inverse,axiom,
    ! [N: nat] : aa_int_nat(nat_int_encode,aa_nat_int(nat_int_decode,N)) = N ).

% int_decode_inverse
tff(fact_8513_inj__int__decode,axiom,
    ! [A4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_int(nat_int_decode),A4)) ).

% inj_int_decode
tff(fact_8514_int__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_int(nat_int_decode,X) = aa_nat_int(nat_int_decode,Y) )
    <=> ( X = Y ) ) ).

% int_decode_eq
tff(fact_8515_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( ( X != zero_zero_real )
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N)) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(power_int_real(X),N) ) ) ) ).

% powr_real_of_int'
tff(fact_8516_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_8517_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                  & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_8518_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_8519_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg4))
    <=> ( ( Deg = Deg4 )
        & ! [X5: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Deg),TreeList),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_8520_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_8521_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_8522_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Deg2),TreeList2),Summary2))),Mima)) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_8523_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] : bit_take_bit_num(zero_zero_nat,M2) = none_num ).

% take_bit_num_simps(1)
tff(fact_8524_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_aak) ).

% Rats_eq_int_div_nat
tff(fact_8525_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_8526_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X)) ) ).

% Rats_no_bot_less
tff(fact_8527_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = sup_sup_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),J),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_8528_min__weak__def,axiom,
    fun_min_weak = sup_su5525570899277871387at_nat(min_ex6901939911449802026at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_8529_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_8530_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_8531_min__strict__def,axiom,
    fun_min_strict = min_ex6901939911449802026at_nat(fun_pair_less) ).

% min_strict_def
tff(fact_8532_max__weak__def,axiom,
    fun_max_weak = sup_su5525570899277871387at_nat(max_ex8135407076693332796at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_8533_max__strict__def,axiom,
    fun_max_strict = max_ex8135407076693332796at_nat(fun_pair_less) ).

% max_strict_def
tff(fact_8534_take__bit__num__def,axiom,
    ! [N: nat,M2: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M2)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M2) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M2)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M2) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M2)))) ) ) ) ).

% take_bit_num_def
tff(fact_8535_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_8536_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_8537_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_8538_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_8539_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M2)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_8540_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_8541_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F5: fun_real_real] :
      ( ! [X4: real] : has_fi5821293074295781190e_real(F,aa_real_real(F5,X4),topolo2177554685111907308n_real(X4,top_top_set_real))
     => ( ! [X4: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F5,X4)))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_8542_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N))) ) ).

% strict_mono_imp_increasing
tff(fact_8543_infinite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N5: nat] : pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N5)),S2)) ) ) ).

% infinite_enumerate
tff(fact_8544_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,aa_int_rat(fract(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).

% positive_rat
tff(fact_8545_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(positive,aa_rat_rat(minus_minus_rat(Y),X))) ) ).

% less_rat_def
tff(fact_8546_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(positive,X))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_8547_Rat_Opositive__def,axiom,
    positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_aal) ).

% Rat.positive_def
tff(fact_8548_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)) ).

% min_Suc_Suc
tff(fact_8549_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_8550_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_8551_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)) ).

% nat_mult_min_right
tff(fact_8552_nat__mult__min__left,axiom,
    ! [M2: nat,N: 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,M2),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_8553_min__diff,axiom,
    ! [M2: nat,I2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)),I2) ).

% min_diff
tff(fact_8554_min__Suc1,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M2) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_aam,N)),M2) ).

% min_Suc1
tff(fact_8555_min__Suc2,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_aan,N)),M2) ).

% min_Suc2
tff(fact_8556_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),less_than))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_than_iff
tff(fact_8557_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_8558_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_8559_pair__less__def,axiom,
    fun_pair_less = lex_prod_nat_nat(less_than,less_than) ).

% pair_less_def
tff(fact_8560_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I2)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,I2),J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,aa_nat_nat(suc,I2)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_8561_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I2)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,I2),J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,aa_nat_nat(suc,I2)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_8562_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K2)),cons_nat(K2,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_8563_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K2)),cons_nat(aa_nat_nat(suc,K2),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_8564_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X4: nat,Xs2: list_nat] : X != cons_nat(X4,Xs2) ) ).

% list_encode.cases
tff(fact_8565_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X4,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_8566_inj__list__encode,axiom,
    ! [A4: set_list_nat] : inj_on_list_nat_nat(nat_list_encode,A4) ).

% inj_list_encode
tff(fact_8567_surj__list__encode,axiom,
    image_list_nat_nat(nat_list_encode,top_top_set_list_nat) = top_top_set_nat ).

% surj_list_encode
tff(fact_8568_list__encode__eq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = aa_list_nat_nat(nat_list_encode,Y) )
    <=> ( X = Y ) ) ).

% list_encode_eq
tff(fact_8569_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_8570_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_8571_upto__aux__rec,axiom,
    ! [J: int,I2: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( upto_aux(I2,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
       => ( upto_aux(I2,J,Js) = upto_aux(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_8572_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa2)))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
               => ( Y = nil_int ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa2))) ) ) ) ).

% upto.pelims
tff(fact_8573_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I2,J)))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
         => ( upto(I2,J) = cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
         => ( upto(I2,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_8574_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2)) ) ).

% upto_Nil
tff(fact_8575_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int = upto(I2,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2)) ) ).

% upto_Nil2
tff(fact_8576_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I2))
     => ( upto(I2,J) = nil_int ) ) ).

% upto_empty
tff(fact_8577_upto__single,axiom,
    ! [I2: int] : upto(I2,I2) = cons_int(I2,nil_int) ).

% upto_single
tff(fact_8578_nth__upto,axiom,
    ! [I2: int,K2: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K2))),J))
     => ( aa_nat_int(nth_int(upto(I2,J)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K2)) ) ) ).

% nth_upto
tff(fact_8579_length__upto,axiom,
    ! [I2: int,J: int] : size_size_list_int(upto(I2,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),I2)),one_one_int)) ).

% length_upto
tff(fact_8580_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M2),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M2),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M2),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_8581_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M2),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_8582_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_8583_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_8584_atLeastAtMost__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I2),J) = set_int2(upto(I2,J)) ).

% atLeastAtMost_upto
tff(fact_8585_distinct__upto,axiom,
    ! [I2: int,J: int] : pp(aa_list_int_bool(distinct_int,upto(I2,J))) ).

% distinct_upto
tff(fact_8586_upto__aux__def,axiom,
    ! [I2: int,J: int,Js: list_int] : upto_aux(I2,J,Js) = append_int(upto(I2,J),Js) ).

% upto_aux_def
tff(fact_8587_upto__code,axiom,
    ! [I2: int,J: int] : upto(I2,J) = upto_aux(I2,J,nil_int) ).

% upto_code
tff(fact_8588_upto__split2,axiom,
    ! [I2: int,J: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K2))
       => ( upto(I2,K2) = append_int(upto(I2,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K2)) ) ) ) ).

% upto_split2
tff(fact_8589_upto__split1,axiom,
    ! [I2: int,J: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K2))
       => ( upto(I2,K2) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K2)) ) ) ) ).

% upto_split1
tff(fact_8590_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I2),J) = set_int2(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_8591_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,I2),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_8592_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( upto(I2,J) = cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_8593_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_8594_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
       => ( upto(I2,J) = cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
       => ( upto(I2,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_8595_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( upto(I2,J) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_8596_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,I2),J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_8597_upto__split3,axiom,
    ! [I2: int,J: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K2))
       => ( upto(I2,K2) = append_int(upto(I2,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),K2))) ) ) ) ).

% upto_split3
tff(fact_8598_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X4,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X4,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_8599_remdups__upt,axiom,
    ! [M2: nat,N: nat] : remdups_nat(upt(M2,N)) = upt(M2,N) ).

% remdups_upt
tff(fact_8600_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( hd_nat(upt(I2,J)) = I2 ) ) ).

% hd_upt
tff(fact_8601_drop__upt,axiom,
    ! [M2: nat,I2: nat,J: nat] : drop_nat(M2,upt(I2,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2),J) ).

% drop_upt
tff(fact_8602_length__upt,axiom,
    ! [I2: nat,J: nat] : size_size_list_nat(upt(I2,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2) ).

% length_upt
tff(fact_8603_take__upt,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2)),N))
     => ( take_nat(M2,upt(I2,N)) = upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2)) ) ) ).

% take_upt
tff(fact_8604_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2))
     => ( upt(I2,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_8605_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),N)) = upt(M2,N) ).

% sorted_list_of_set_range
tff(fact_8606_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( upt(I2,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I2)) ) ) ).

% upt_eq_Nil_conv
tff(fact_8607_nth__upt,axiom,
    ! [I2: nat,K2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J))
     => ( aa_nat_nat(nth_nat(upt(I2,J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) ) ) ).

% nth_upt
tff(fact_8608_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M2),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M2),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M2),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_8609_upt__0,axiom,
    ! [I2: nat] : upt(I2,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_8610_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M2,cons_nat(N,Ns)) = upt(M2,Q2) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M2),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_8611_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : map_nat_nat(suc,upt(M2,N)) = upt(aa_nat_nat(suc,M2),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_8612_map__add__upt,axiom,
    ! [N: nat,M2: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aao,N),upt(zero_zero_nat,M2)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% map_add_upt
tff(fact_8613_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or5834768355832116004an_nat,N),M2) = set_nat2(upt(aa_nat_nat(suc,N),M2)) ).

% greaterThanLessThan_upt
tff(fact_8614_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or6659071591806873216st_nat,N),M2) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M2))) ).

% greaterThanAtMost_upt
tff(fact_8615_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_8616_atLeastLessThan__upt,axiom,
    ! [I2: nat,J: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I2),J) = set_nat2(upt(I2,J)) ).

% atLeastLessThan_upt
tff(fact_8617_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M2) = set_nat2(upt(N,aa_nat_nat(suc,M2))) ).

% atLeastAtMost_upt
tff(fact_8618_distinct__upt,axiom,
    ! [I2: nat,J: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I2,J))) ).

% distinct_upt
tff(fact_8619_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_8620_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
     => ( upt(I2,J) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J)) ) ) ).

% upt_conv_Cons
tff(fact_8621_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : map_nat_nat(aTP_Lamm_aap,upt(aa_nat_nat(suc,M2),aa_nat_nat(suc,N))) = upt(M2,N) ).

% map_decr_upt
tff(fact_8622_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = append_nat(upt(I2,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% upt_add_eq_append
tff(fact_8623_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I2,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
        & ( I2 = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_8624_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
       => ( upt(I2,J) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
       => ( upt(I2,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_8625_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
       => ( upt(I2,aa_nat_nat(suc,J)) = append_nat(upt(I2,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
       => ( upt(I2,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_8626_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J))
     => ( upt(I2,aa_nat_nat(suc,J)) = append_nat(upt(I2,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_8627_sum__list__upt,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups4561878855575611511st_nat(upt(M2,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M2),N)) ) ) ).

% sum_list_upt
tff(fact_8628_card__length__sum__list__rec,axiom,
    ! [M2: nat,N4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M2))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_aaq(M2),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_aar(M2),N4)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_aas(M2),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_8629_card__length__sum__list,axiom,
    ! [M2: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_aaq(M2),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),M2)),one_one_nat)),N4) ).

% card_length_sum_list
tff(fact_8630_sorted__upt,axiom,
    ! [M2: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M2,N))) ).

% sorted_upt
tff(fact_8631_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M2,N))) ).

% sorted_wrt_upt
tff(fact_8632_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(nth_nat(Ns),I2))) ) ) ).

% sorted_wrt_less_idx
tff(fact_8633_sorted__upto,axiom,
    ! [M2: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M2,N))) ).

% sorted_upto
tff(fact_8634_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I2,J))) ).

% sorted_wrt_upto
tff(fact_8635_tl__upt,axiom,
    ! [M2: nat,N: nat] : tl_nat(upt(M2,N)) = upt(aa_nat_nat(suc,M2),N) ).

% tl_upt
tff(fact_8636_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uy(M2))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M2),aa_nat1818925681583257441et_nat(aTP_Lamm_aat,M2)) ).

% pairs_le_eq_Sigma
tff(fact_8637_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_8638_vimage__Suc__insert__0,axiom,
    ! [A4: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A4)) = vimage_nat_nat(suc,A4) ).

% vimage_Suc_insert_0
tff(fact_8639_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A4: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,N),A4)) = insert_nat(N,vimage_nat_nat(suc,A4)) ).

% vimage_Suc_insert_Suc
tff(fact_8640_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,vimage_nat_nat(suc,F3)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,F3)) ) ).

% finite_vimage_Suc_iff
tff(fact_8641_set__decode__div__2,axiom,
    ! [X: nat] : nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,nat_set_decode(X)) ).

% set_decode_div_2
tff(fact_8642_set__encode__vimage__Suc,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A4)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_8643_Restr__natLeq,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,N)),aa_nat1818925681583257441et_nat(aTP_Lamm_aau,N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_aav(N))) ).

% Restr_natLeq
tff(fact_8644_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_8645_Restr__natLeq2,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,N),aa_nat1818925681583257441et_nat(aTP_Lamm_aaw,N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_aav(N))) ).

% Restr_natLeq2
tff(fact_8646_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,N)) ).

% natLeq_underS_less
tff(fact_8647_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ).

% Arg_bounded
tff(fact_8648_total__pair__less,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : total_3592101749530773125at_nat(A4,fun_pair_less) ).

% total_pair_less
tff(fact_8649_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ) ).

% Arg_correct
tff(fact_8650_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2)) ) ).

% not_negative_int_iff
tff(fact_8651_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_8652_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => ( aa_complex_real(arg,Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_8653_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_aax(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_dc,N)))) ) ).

% bij_betw_roots_unity
tff(fact_8654_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_set_complex_bool(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,N),aa_complex_real(real_V1022390504157884413omplex,C2)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C2)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_dc,N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_ye(C2),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_8655_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_real(arg,Z) = zero_zero_real ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_real(arg,Z) = fChoice_real(aTP_Lamm_aay(Z)) ) ) ) ).

% Arg_def
tff(fact_8656_bij__betw__Suc,axiom,
    ! [M5: set_nat,N4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(suc),M5),N4))
    <=> ( image_nat_nat(suc,M5) = N4 ) ) ).

% bij_betw_Suc
tff(fact_8657_bij__int__decode,axiom,
    pp(aa_set_int_bool(bij_betw_nat_int(nat_int_decode,top_top_set_nat),top_top_set_int)) ).

% bij_int_decode
tff(fact_8658_bij__int__encode,axiom,
    pp(aa_set_nat_bool(bij_betw_int_nat(nat_int_encode,top_top_set_int),top_top_set_nat)) ).

% bij_int_encode
tff(fact_8659_bij__list__encode,axiom,
    pp(aa_set_nat_bool(bij_be8532844293280997160at_nat(nat_list_encode,top_top_set_list_nat),top_top_set_nat)) ).

% bij_list_encode
tff(fact_8660_bij__prod__encode,axiom,
    pp(aa_set_nat_bool(bij_be5333170631980326235at_nat(nat_prod_encode,top_to4669805908274784177at_nat),top_top_set_nat)) ).

% bij_prod_encode
tff(fact_8661_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_gcd_int,Xs),zero_zero_int) ).

% Gcd_int_set_eq_fold
tff(fact_8662_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_8663_sort__upt,axiom,
    ! [M2: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_ce,upt(M2,N)) = upt(M2,N) ).

% sort_upt
tff(fact_8664_sort__upto,axiom,
    ! [I2: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_cf,upto(I2,J)) = upto(I2,J) ).

% sort_upto
tff(fact_8665_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_aav(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,N)) ).

% Field_natLeq_on
tff(fact_8666_pair__leq__def,axiom,
    fun_pair_leq = sup_su718114333110466843at_nat(fun_pair_less,id_Pro2258643101195443293at_nat) ).

% pair_leq_def
tff(fact_8667_wf__less,axiom,
    pp(aa_set3135850010578026034t_bool(wf_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)))) ).

% wf_less
tff(fact_8668_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat(fun_pair_less) ).

% wf_pair_less
tff(fact_8669_wf__int__ge__less__than2,axiom,
    ! [D: int] : wf_int(int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_8670_wf__int__ge__less__than,axiom,
    ! [D: int] : wf_int(int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_8671_cauchyD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( cauchy(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K: nat] :
          ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M4))
           => ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M4)),aa_nat_rat(X6,N5)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_8672_cauchyI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K4: nat] :
            ! [M: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),M))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M)),aa_nat_rat(X6,N3)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_8673_cauchy__imp__bounded,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ? [B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
          & ! [N5: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),B3)) ) ) ).

% cauchy_imp_bounded
tff(fact_8674_cauchy__def,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M3))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X6,M3)),aa_nat_rat(X6,N2)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_8675_le__Real,axiom,
    ! [X6: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real2(X6)),real2(Y6)))
        <=> ! [R5: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
             => ? [K3: nat] :
                ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N2)),aa_rat_rat(plus_plus_rat(aa_nat_rat(Y6,N2)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_8676_cauchy__not__vanishes,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K: nat] :
              ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_8677_vanishes__mult__bounded,axiom,
    ! [X6: fun_nat_rat,Y6: fun_nat_rat] :
      ( ? [A8: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A8))
          & ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),A8)) )
     => ( vanishes(Y6)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_aaz(X6),Y6)) ) ) ).

% vanishes_mult_bounded
tff(fact_8678_vanishes__def,axiom,
    ! [X6: fun_nat_rat] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N2))),R5)) ) ) ) ).

% vanishes_def
tff(fact_8679_vanishesI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K4: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),R3)) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_8680_vanishesD,axiom,
    ! [X6: fun_nat_rat,R2: rat] :
      ( vanishes(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N5))),R2)) ) ) ) ).

% vanishesD
tff(fact_8681_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K: nat] :
                ( ! [N5: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X6,N5)))) )
                | ! [N5: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_nat_rat(X6,N5))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_8682_not__positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ pp(aa_real_bool(positive2,real2(X6)))
      <=> ! [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
           => ? [K3: nat] :
              ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N2)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_8683_positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( pp(aa_real_bool(positive2,real2(X6)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X6,N2))) ) ) ) ) ).

% positive_Real
tff(fact_8684_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X))) ) ).

% less_real_def
tff(fact_8685_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X),N2))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_8686_Real_Opositive__def,axiom,
    positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_aba) ).

% Real.positive_def
tff(fact_8687_le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ).

% le_enumerate
tff(fact_8688_enumerate__Ex,axiom,
    ! [S2: set_nat,S: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(member_nat(S),S2))
       => ? [N3: nat] : aa_nat_nat(infini8530281810654367211te_nat(S2),N3) = S ) ) ).

% enumerate_Ex
tff(fact_8689_strict__mono__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => order_5726023648592871131at_nat(infini8530281810654367211te_nat(S2)) ) ).

% strict_mono_enumerate
tff(fact_8690_range__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( image_nat_nat(infini8530281810654367211te_nat(S2),top_top_set_nat) = S2 ) ) ).

% range_enumerate
tff(fact_8691_finite__le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ) ).

% finite_le_enumerate
tff(fact_8692_bij__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(bij_betw_nat_nat(infini8530281810654367211te_nat(S2)),top_top_set_nat),S2)) ) ).

% bij_enumerate
tff(fact_8693_Least__eq__0,axiom,
    ! [P2: fun_nat_bool] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ord_Least_nat(P2) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_8694_Least__Suc,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
       => ( ord_Least_nat(P2) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_abb(P2))) ) ) ) ).

% Least_Suc
tff(fact_8695_Least__Suc2,axiom,
    ! [P2: fun_nat_bool,N: nat,Q: fun_nat_bool,M2: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( pp(aa_nat_bool(Q,M2))
       => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
         => ( ! [K: nat] :
                ( pp(aa_nat_bool(P2,aa_nat_nat(suc,K)))
              <=> pp(aa_nat_bool(Q,K)) )
           => ( ord_Least_nat(P2) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_8696_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_op,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_8697_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aap,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_2
tff(fact_8698_ATP_Olambda__3,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_my,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_3
tff(fact_8699_ATP_Olambda__4,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_nb,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_4
tff(fact_8700_ATP_Olambda__5,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_mz,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_5
tff(fact_8701_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_nc,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_6
tff(fact_8702_ATP_Olambda__7,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_na,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_7
tff(fact_8703_ATP_Olambda__8,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_aal,Uu))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu)))) ) ).

% ATP.lambda_8
tff(fact_8704_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_zo,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xr,Uu)) ).

% ATP.lambda_9
tff(fact_8705_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_zq,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf,Uu)) ).

% ATP.lambda_10
tff(fact_8706_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_aah,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_aag(Uu)))) ).

% ATP.lambda_11
tff(fact_8707_ATP_Olambda__12,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_aaf,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_aae(Uu)))) ).

% ATP.lambda_12
tff(fact_8708_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_fm,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_13
tff(fact_8709_ATP_Olambda__14,axiom,
    ! [Uu: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_aba,Uu))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu,N2))) ) ) ) ).

% ATP.lambda_14
tff(fact_8710_ATP_Olambda__15,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_aak,Uu))
    <=> ? [I4: int,N2: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
          & ( N2 != zero_zero_nat ) ) ) ).

% ATP.lambda_15
tff(fact_8711_ATP_Olambda__16,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_uc(Uu),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_16
tff(fact_8712_ATP_Olambda__17,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_uf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_17
tff(fact_8713_ATP_Olambda__18,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ud(Uu),Uua) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_rat) ).

% ATP.lambda_18
tff(fact_8714_ATP_Olambda__19,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ue,Uu),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_19
tff(fact_8715_ATP_Olambda__20,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_mn,Uu),Uua) = if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),one_on7984719198319812577d_enat)) ).

% ATP.lambda_20
tff(fact_8716_ATP_Olambda__21,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ml,Uu),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),one_one_complex)) ).

% ATP.lambda_21
tff(fact_8717_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu))),one_one_real)) ).

% ATP.lambda_22
tff(fact_8718_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mm,Uu),Uua) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu))),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu))),one_one_rat)) ).

% ATP.lambda_23
tff(fact_8719_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mq,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu))),one_one_nat)) ).

% ATP.lambda_24
tff(fact_8720_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mo,Uu),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int)) ).

% ATP.lambda_25
tff(fact_8721_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ty(Uu),Uua) = aa_complex_complex(if_complex(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_26
tff(fact_8722_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ub,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_27
tff(fact_8723_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_tz(Uu),Uua) = aa_rat_rat(if_rat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_rat) ).

% ATP.lambda_28
tff(fact_8724_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ua,Uu),Uua) = aa_int_int(if_int(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_29
tff(fact_8725_ATP_Olambda__30,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ve(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_vd(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_30
tff(fact_8726_ATP_Olambda__31,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_vf(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_31
tff(fact_8727_ATP_Olambda__32,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vc(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vb(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_32
tff(fact_8728_ATP_Olambda__33,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_va(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_33
tff(fact_8729_ATP_Olambda__34,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_34
tff(fact_8730_ATP_Olambda__35,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_35
tff(fact_8731_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xa(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_36
tff(fact_8732_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xd,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_37
tff(fact_8733_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xb(Uu),Uua) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_38
tff(fact_8734_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xc,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_39
tff(fact_8735_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_xy,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_40
tff(fact_8736_ATP_Olambda__41,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aay(Uu),Uua))
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi)) ) ) ).

% ATP.lambda_41
tff(fact_8737_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_xz,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_42
tff(fact_8738_ATP_Olambda__43,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ya(Uu),Uua))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_43
tff(fact_8739_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_44
tff(fact_8740_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_45
tff(fact_8741_ATP_Olambda__46,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_46
tff(fact_8742_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xm(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_47
tff(fact_8743_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_48
tff(fact_8744_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xn(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_49
tff(fact_8745_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xo,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_50
tff(fact_8746_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_yj,Uu),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_51
tff(fact_8747_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xs,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_52
tff(fact_8748_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aai,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_53
tff(fact_8749_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ux,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_54
tff(fact_8750_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uw,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_55
tff(fact_8751_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_dc,Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_56
tff(fact_8752_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_db,Uu),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_57
tff(fact_8753_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xw,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_58
tff(fact_8754_ATP_Olambda__59,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sj(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,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_59
tff(fact_8755_ATP_Olambda__60,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_60
tff(fact_8756_ATP_Olambda__61,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sl(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_61
tff(fact_8757_ATP_Olambda__62,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_sn(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,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_62
tff(fact_8758_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sm(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,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_63
tff(fact_8759_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lz(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_64
tff(fact_8760_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lx(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_65
tff(fact_8761_ATP_Olambda__66,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ly(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_66
tff(fact_8762_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lw(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_67
tff(fact_8763_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_le(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_68
tff(fact_8764_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ld(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_69
tff(fact_8765_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lc(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_70
tff(fact_8766_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_jr(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_71
tff(fact_8767_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_72
tff(fact_8768_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_js(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_73
tff(fact_8769_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lv(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_74
tff(fact_8770_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lu(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_75
tff(fact_8771_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lt(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_76
tff(fact_8772_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lp(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_77
tff(fact_8773_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ls(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_78
tff(fact_8774_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lr(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_79
tff(fact_8775_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lq(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_80
tff(fact_8776_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nm(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_81
tff(fact_8777_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nl(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_82
tff(fact_8778_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nk(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_83
tff(fact_8779_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uu(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_84
tff(fact_8780_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_us(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_85
tff(fact_8781_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_up(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_86
tff(fact_8782_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_um(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_87
tff(fact_8783_ATP_Olambda__88,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_jd(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_88
tff(fact_8784_ATP_Olambda__89,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_jg(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_89
tff(fact_8785_ATP_Olambda__90,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_iz(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_90
tff(fact_8786_ATP_Olambda__91,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_jb(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_91
tff(fact_8787_ATP_Olambda__92,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_je(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_92
tff(fact_8788_ATP_Olambda__93,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_jh(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_93
tff(fact_8789_ATP_Olambda__94,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ja(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_94
tff(fact_8790_ATP_Olambda__95,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_jc(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_95
tff(fact_8791_ATP_Olambda__96,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_jf(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_96
tff(fact_8792_ATP_Olambda__97,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ji(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_97
tff(fact_8793_ATP_Olambda__98,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_rq(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_98
tff(fact_8794_ATP_Olambda__99,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_rl(Uu),Uua))
    <=> ( aa_complex_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_99
tff(fact_8795_ATP_Olambda__100,axiom,
    ! [Uu: fun_complex_int,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_rn(Uu),Uua))
    <=> ( aa_complex_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_100
tff(fact_8796_ATP_Olambda__101,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_rr(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_101
tff(fact_8797_ATP_Olambda__102,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ro(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_102
tff(fact_8798_ATP_Olambda__103,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_rj(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_103
tff(fact_8799_ATP_Olambda__104,axiom,
    ! [Uu: fun_real_int,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_rm(Uu),Uua))
    <=> ( aa_real_int(Uu,Uua) = one_one_int ) ) ).

% ATP.lambda_104
tff(fact_8800_ATP_Olambda__105,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_rs(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_105
tff(fact_8801_ATP_Olambda__106,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_rp(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_106
tff(fact_8802_ATP_Olambda__107,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_rk(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_107
tff(fact_8803_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aaa(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_108
tff(fact_8804_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zz(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zt(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_109
tff(fact_8805_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yv,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_110
tff(fact_8806_ATP_Olambda__111,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_su(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_111
tff(fact_8807_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_st,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_112
tff(fact_8808_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_113
tff(fact_8809_ATP_Olambda__114,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_114
tff(fact_8810_ATP_Olambda__115,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xx,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_115
tff(fact_8811_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ew,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_116
tff(fact_8812_ATP_Olambda__117,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ev,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_117
tff(fact_8813_ATP_Olambda__118,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_eu,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_118
tff(fact_8814_ATP_Olambda__119,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_et,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_119
tff(fact_8815_ATP_Olambda__120,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_om(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_120
tff(fact_8816_ATP_Olambda__121,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oo,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_121
tff(fact_8817_ATP_Olambda__122,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_on(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_122
tff(fact_8818_ATP_Olambda__123,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_123
tff(fact_8819_ATP_Olambda__124,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sb(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_124
tff(fact_8820_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_se,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_125
tff(fact_8821_ATP_Olambda__126,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_sd(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_126
tff(fact_8822_ATP_Olambda__127,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ug,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).

% ATP.lambda_127
tff(fact_8823_ATP_Olambda__128,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(aTP_Lamm_ax,Uu),Uua))
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,Uua),Uu)) ) ).

% ATP.lambda_128
tff(fact_8824_ATP_Olambda__129,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_aw(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_129
tff(fact_8825_ATP_Olambda__130,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_av,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Uua),Uu)) ) ).

% ATP.lambda_130
tff(fact_8826_ATP_Olambda__131,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_ay(Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_131
tff(fact_8827_ATP_Olambda__132,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_uh,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_132
tff(fact_8828_ATP_Olambda__133,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_ui,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).

% ATP.lambda_133
tff(fact_8829_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bc,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_134
tff(fact_8830_ATP_Olambda__135,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_uj,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_135
tff(fact_8831_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_136
tff(fact_8832_ATP_Olambda__137,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_nh,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_137
tff(fact_8833_ATP_Olambda__138,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_nd,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,Uua),Uu)) ) ).

% ATP.lambda_138
tff(fact_8834_ATP_Olambda__139,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_ne,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_139
tff(fact_8835_ATP_Olambda__140,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_nf,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).

% ATP.lambda_140
tff(fact_8836_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bb,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_8837_ATP_Olambda__142,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ng,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_142
tff(fact_8838_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_143
tff(fact_8839_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_144
tff(fact_8840_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_za,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_145
tff(fact_8841_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aao,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_146
tff(fact_8842_ATP_Olambda__147,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_yx(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_147
tff(fact_8843_ATP_Olambda__148,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_zb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_148
tff(fact_8844_ATP_Olambda__149,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_be,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_8845_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bf,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_150
tff(fact_8846_ATP_Olambda__151,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bd,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_151
tff(fact_8847_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ym,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_152
tff(fact_8848_ATP_Olambda__153,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zw,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_153
tff(fact_8849_ATP_Olambda__154,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uua),Uu)) ) ).

% ATP.lambda_154
tff(fact_8850_ATP_Olambda__155,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aa(Uu),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_155
tff(fact_8851_ATP_Olambda__156,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_156
tff(fact_8852_ATP_Olambda__157,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_157
tff(fact_8853_ATP_Olambda__158,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ad(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_158
tff(fact_8854_ATP_Olambda__159,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_159
tff(fact_8855_ATP_Olambda__160,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aTP_Lamm_jk(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_160
tff(fact_8856_ATP_Olambda__161,axiom,
    ! [Uu: product_prod_nat_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_jn,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_161
tff(fact_8857_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jo,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_162
tff(fact_8858_ATP_Olambda__163,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_jl(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_163
tff(fact_8859_ATP_Olambda__164,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_jm,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_164
tff(fact_8860_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_jp,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_165
tff(fact_8861_ATP_Olambda__166,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jq,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_166
tff(fact_8862_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_aaw,Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_167
tff(fact_8863_ATP_Olambda__168,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oj(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_168
tff(fact_8864_ATP_Olambda__169,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oi(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_169
tff(fact_8865_ATP_Olambda__170,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rv(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_170
tff(fact_8866_ATP_Olambda__171,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kr(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_171
tff(fact_8867_ATP_Olambda__172,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_abb(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_172
tff(fact_8868_ATP_Olambda__173,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lb(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_173
tff(fact_8869_ATP_Olambda__174,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_kq(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_174
tff(fact_8870_ATP_Olambda__175,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_la(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_175
tff(fact_8871_ATP_Olambda__176,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_yu,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yt(Uu),Uua)) ).

% ATP.lambda_176
tff(fact_8872_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ys,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yr(Uu),Uua)) ).

% ATP.lambda_177
tff(fact_8873_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_yq,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yp(Uu),Uua)) ).

% ATP.lambda_178
tff(fact_8874_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_yo,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yn(Uu),Uua)) ).

% ATP.lambda_179
tff(fact_8875_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_yl,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yk(Uu),Uua)) ).

% ATP.lambda_180
tff(fact_8876_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_zg(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_zf(Uu),Uua)) ).

% ATP.lambda_181
tff(fact_8877_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_aax(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_182
tff(fact_8878_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cn,Uu),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_183
tff(fact_8879_ATP_Olambda__184,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_oq(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_184
tff(fact_8880_ATP_Olambda__185,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ot(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_185
tff(fact_8881_ATP_Olambda__186,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_or(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_186
tff(fact_8882_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ou(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_187
tff(fact_8883_ATP_Olambda__188,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_os(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_188
tff(fact_8884_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_aat,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_189
tff(fact_8885_ATP_Olambda__190,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zy,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_190
tff(fact_8886_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aam,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_191
tff(fact_8887_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aan,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_192
tff(fact_8888_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yh,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_193
tff(fact_8889_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yg,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_194
tff(fact_8890_ATP_Olambda__195,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zv(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_195
tff(fact_8891_ATP_Olambda__196,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_md(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_196
tff(fact_8892_ATP_Olambda__197,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_mg(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_197
tff(fact_8893_ATP_Olambda__198,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mf(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_198
tff(fact_8894_ATP_Olambda__199,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_me(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_199
tff(fact_8895_ATP_Olambda__200,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_mc(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_200
tff(fact_8896_ATP_Olambda__201,axiom,
    ! [Uu: fun_Pr672294278902479040t_real,Uua: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aTP_Lamm_tw(Uu),Uua) = aa_real_real(ln_ln_real,aa_Pro6168199777849092593t_real(Uu,Uua)) ).

% ATP.lambda_201
tff(fact_8897_ATP_Olambda__202,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_tv(Uu),Uua) = aa_real_real(ln_ln_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_202
tff(fact_8898_ATP_Olambda__203,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ts(Uu),Uua) = aa_real_real(ln_ln_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_203
tff(fact_8899_ATP_Olambda__204,axiom,
    ! [Uu: fun_set_nat_real,Uua: set_nat] : aa_set_nat_real(aTP_Lamm_tt(Uu),Uua) = aa_real_real(ln_ln_real,aa_set_nat_real(Uu,Uua)) ).

% ATP.lambda_204
tff(fact_8900_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tx(Uu),Uua) = aa_real_real(ln_ln_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_205
tff(fact_8901_ATP_Olambda__206,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_tu(Uu),Uua) = aa_real_real(ln_ln_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_206
tff(fact_8902_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_qt(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_8903_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_qu(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_8904_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_qv(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_8905_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_qx(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_8906_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_qw(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_8907_ATP_Olambda__212,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_gm(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_8908_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_gn(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_8909_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_ov(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_8910_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ow(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_8911_ATP_Olambda__216,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_go(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_8912_ATP_Olambda__217,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_gp(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_8913_ATP_Olambda__218,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_gq(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_8914_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_so(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_8915_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ss(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_8916_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_sp(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_8917_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_sr(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_8918_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_sq(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_8919_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xu(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_8920_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_aau,Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bb,Uu)) ).

% ATP.lambda_225
tff(fact_8921_ATP_Olambda__226,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_226
tff(fact_8922_ATP_Olambda__227,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_es(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_227
tff(fact_8923_ATP_Olambda__228,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_er(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_228
tff(fact_8924_ATP_Olambda__229,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tj(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_229
tff(fact_8925_ATP_Olambda__230,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_230
tff(fact_8926_ATP_Olambda__231,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_231
tff(fact_8927_ATP_Olambda__232,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_tk(Uu),Uua),Uub) = aa_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_232
tff(fact_8928_ATP_Olambda__233,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_gh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_233
tff(fact_8929_ATP_Olambda__234,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gl(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_234
tff(fact_8930_ATP_Olambda__235,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_gc(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_235
tff(fact_8931_ATP_Olambda__236,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_236
tff(fact_8932_ATP_Olambda__237,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gi(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_237
tff(fact_8933_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gd(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_238
tff(fact_8934_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gj(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_239
tff(fact_8935_ATP_Olambda__240,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ge(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_240
tff(fact_8936_ATP_Olambda__241,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_gg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_241
tff(fact_8937_ATP_Olambda__242,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_gk(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_242
tff(fact_8938_ATP_Olambda__243,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_pf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_243
tff(fact_8939_ATP_Olambda__244,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_oz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_244
tff(fact_8940_ATP_Olambda__245,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_pb(Uu),Uua),Uub) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_245
tff(fact_8941_ATP_Olambda__246,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_pg(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_246
tff(fact_8942_ATP_Olambda__247,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_247
tff(fact_8943_ATP_Olambda__248,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ox(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_248
tff(fact_8944_ATP_Olambda__249,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_pa(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_249
tff(fact_8945_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_250
tff(fact_8946_ATP_Olambda__251,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_pe(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_251
tff(fact_8947_ATP_Olambda__252,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_oy(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_252
tff(fact_8948_ATP_Olambda__253,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_253
tff(fact_8949_ATP_Olambda__254,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gb(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_254
tff(fact_8950_ATP_Olambda__255,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_fs(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_255
tff(fact_8951_ATP_Olambda__256,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_256
tff(fact_8952_ATP_Olambda__257,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fy(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_257
tff(fact_8953_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ft(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_258
tff(fact_8954_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_259
tff(fact_8955_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fz(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_260
tff(fact_8956_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_sw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_261
tff(fact_8957_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sx(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_262
tff(fact_8958_ATP_Olambda__263,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_fu(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_263
tff(fact_8959_ATP_Olambda__264,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_264
tff(fact_8960_ATP_Olambda__265,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ga(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_265
tff(fact_8961_ATP_Olambda__266,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_pp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_266
tff(fact_8962_ATP_Olambda__267,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_pj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_267
tff(fact_8963_ATP_Olambda__268,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_pl(Uu),Uua),Uub) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_268
tff(fact_8964_ATP_Olambda__269,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_pq(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_269
tff(fact_8965_ATP_Olambda__270,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_270
tff(fact_8966_ATP_Olambda__271,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ph(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_271
tff(fact_8967_ATP_Olambda__272,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_pk(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_int(Uua,Uub)),one_one_int) ).

% ATP.lambda_272
tff(fact_8968_ATP_Olambda__273,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_273
tff(fact_8969_ATP_Olambda__274,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_po(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_274
tff(fact_8970_ATP_Olambda__275,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pi(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_nat(Uua,Uub)),one_one_nat) ).

% ATP.lambda_275
tff(fact_8971_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tf(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uu,Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_276
tff(fact_8972_ATP_Olambda__277,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ti(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uu,Uub)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_277
tff(fact_8973_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_th(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uu,Uub)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_278
tff(fact_8974_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_tg(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_279
tff(fact_8975_ATP_Olambda__280,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_iu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_280
tff(fact_8976_ATP_Olambda__281,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_bool,Uub: complex] : aa_complex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_iy(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(Uua,Uub),aa_complex_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_281
tff(fact_8977_ATP_Olambda__282,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_im(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_282
tff(fact_8978_ATP_Olambda__283,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_ir(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_283
tff(fact_8979_ATP_Olambda__284,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_iv(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_284
tff(fact_8980_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_io(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_285
tff(fact_8981_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_iw(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(Uua,Uub),aa_nat_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_286
tff(fact_8982_ATP_Olambda__287,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_iq(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_287
tff(fact_8983_ATP_Olambda__288,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_is(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_288
tff(fact_8984_ATP_Olambda__289,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_bool,Uub: int] : aa_int_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_ix(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(Uua,Uub),aa_int_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_289
tff(fact_8985_ATP_Olambda__290,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_rg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_290
tff(fact_8986_ATP_Olambda__291,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_bool,Uub: complex] : aa_complex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_ra(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(Uua,Uub)),aa_complex_nat(Uu,Uub)),one_one_nat) ).

% ATP.lambda_291
tff(fact_8987_ATP_Olambda__292,axiom,
    ! [Uu: fun_complex_int,Uua: fun_complex_bool,Uub: complex] : aa_complex_int(aa_fun4860376958843682909ex_int(aTP_Lamm_rc(Uu),Uua),Uub) = aa_int_int(if_int(aa_complex_bool(Uua,Uub),aa_complex_int(Uu,Uub)),one_one_int) ).

% ATP.lambda_292
tff(fact_8988_ATP_Olambda__293,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_rh(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_293
tff(fact_8989_ATP_Olambda__294,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_rd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_294
tff(fact_8990_ATP_Olambda__295,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_bool,Uub: real] : aa_real_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_qy(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(Uua,Uub)),aa_real_nat(Uu,Uub)),one_one_nat) ).

% ATP.lambda_295
tff(fact_8991_ATP_Olambda__296,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_bool,Uub: real] : aa_real_int(aa_fun6550077132417316957al_int(aTP_Lamm_rb(Uu),Uua),Uub) = aa_int_int(if_int(aa_real_bool(Uua,Uub),aa_real_int(Uu,Uub)),one_one_int) ).

% ATP.lambda_296
tff(fact_8992_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_bool,Uub: nat] : aa_nat_real(aa_fun1750223370215434333t_real(aTP_Lamm_re(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uua,Uub)),aa_nat_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_297
tff(fact_8993_ATP_Olambda__298,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_rf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_298
tff(fact_8994_ATP_Olambda__299,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_bool,Uub: int] : aa_int_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_qz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uua,Uub)),aa_int_nat(Uu,Uub)),one_one_nat) ).

% ATP.lambda_299
tff(fact_8995_ATP_Olambda__300,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vb(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_300
tff(fact_8996_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uz(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_301
tff(fact_8997_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_vd(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_302
tff(fact_8998_ATP_Olambda__303,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hu(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_303
tff(fact_8999_ATP_Olambda__304,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_pv(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_304
tff(fact_9000_ATP_Olambda__305,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_hh(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_305
tff(fact_9001_ATP_Olambda__306,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_hy(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_306
tff(fact_9002_ATP_Olambda__307,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ij(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_307
tff(fact_9003_ATP_Olambda__308,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_hm(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_308
tff(fact_9004_ATP_Olambda__309,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_ps(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_309
tff(fact_9005_ATP_Olambda__310,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_gz(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_310
tff(fact_9006_ATP_Olambda__311,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ic(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_311
tff(fact_9007_ATP_Olambda__312,axiom,
    ! [Uu: fun_na5967208032043182372ex_int,Uua: complex,Uub: nat] : aa_nat_int(aa_com2205869648058525157at_int(aTP_Lamm_qn(Uu),Uua),Uub) = aa_complex_int(aa_nat8436805559129492965ex_int(Uu,Uub),Uua) ).

% ATP.lambda_312
tff(fact_9008_ATP_Olambda__313,axiom,
    ! [Uu: fun_nat_fun_real_int,Uua: real,Uub: nat] : aa_nat_int(aa_real_fun_nat_int(aTP_Lamm_qj(Uu),Uua),Uub) = aa_real_int(aa_nat_fun_real_int(Uu,Uub),Uua) ).

% ATP.lambda_313
tff(fact_9009_ATP_Olambda__314,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_uq(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_314
tff(fact_9010_ATP_Olambda__315,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_un(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_315
tff(fact_9011_ATP_Olambda__316,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qr(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_316
tff(fact_9012_ATP_Olambda__317,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hd(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_317
tff(fact_9013_ATP_Olambda__318,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_ig(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_318
tff(fact_9014_ATP_Olambda__319,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_hq(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_319
tff(fact_9015_ATP_Olambda__320,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tn(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_complex),zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_320
tff(fact_9016_ATP_Olambda__321,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_to(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_real),zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_321
tff(fact_9017_ATP_Olambda__322,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_tp(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_int),zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_322
tff(fact_9018_ATP_Olambda__323,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uv(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_un(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_323
tff(fact_9019_ATP_Olambda__324,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ut(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qr(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_324
tff(fact_9020_ATP_Olambda__325,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ur(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_uq(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_325
tff(fact_9021_ATP_Olambda__326,axiom,
    ! [Uu: fun_nat_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(aa_nat_fun_nat_nat(aTP_Lamm_un(Uu),Uub)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_326
tff(fact_9022_ATP_Olambda__327,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_327
tff(fact_9023_ATP_Olambda__328,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_zi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_328
tff(fact_9024_ATP_Olambda__329,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_vj(Uu),Uua),Uub))
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ul(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_329
tff(fact_9025_ATP_Olambda__330,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_vi(Uu),Uua),Uub))
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_uk(Uu),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_330
tff(fact_9026_ATP_Olambda__331,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ju(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_331
tff(fact_9027_ATP_Olambda__332,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_332
tff(fact_9028_ATP_Olambda__333,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_jv(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_333
tff(fact_9029_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ze(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_334
tff(fact_9030_ATP_Olambda__335,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_335
tff(fact_9031_ATP_Olambda__336,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_aar(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_336
tff(fact_9032_ATP_Olambda__337,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_aas(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_337
tff(fact_9033_ATP_Olambda__338,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cw(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua)) ) ) ).

% ATP.lambda_338
tff(fact_9034_ATP_Olambda__339,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_cv(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_339
tff(fact_9035_ATP_Olambda__340,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_cu(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).

% ATP.lambda_340
tff(fact_9036_ATP_Olambda__341,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cx(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Uub)),Uua)) ) ) ).

% ATP.lambda_341
tff(fact_9037_ATP_Olambda__342,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_cy(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_342
tff(fact_9038_ATP_Olambda__343,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_cz(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Uub)),Uua)) ) ) ).

% ATP.lambda_343
tff(fact_9039_ATP_Olambda__344,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cq(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_344
tff(fact_9040_ATP_Olambda__345,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Uub)),Uu))
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_345
tff(fact_9041_ATP_Olambda__346,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_co(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_346
tff(fact_9042_ATP_Olambda__347,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cr(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_347
tff(fact_9043_ATP_Olambda__348,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_cs(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Uub)),Uu))
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_348
tff(fact_9044_ATP_Olambda__349,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_ct(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_349
tff(fact_9045_ATP_Olambda__350,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_aaq(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_350
tff(fact_9046_ATP_Olambda__351,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_yc(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_351
tff(fact_9047_ATP_Olambda__352,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_yi(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(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_352
tff(fact_9048_ATP_Olambda__353,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aav(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_353
tff(fact_9049_ATP_Olambda__354,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_yd(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_354
tff(fact_9050_ATP_Olambda__355,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vg(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_355
tff(fact_9051_ATP_Olambda__356,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fn(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_356
tff(fact_9052_ATP_Olambda__357,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_at(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_357
tff(fact_9053_ATP_Olambda__358,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fo(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_358
tff(fact_9054_ATP_Olambda__359,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ba(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_359
tff(fact_9055_ATP_Olambda__360,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_az(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_360
tff(fact_9056_ATP_Olambda__361,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_au(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_361
tff(fact_9057_ATP_Olambda__362,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zs(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).

% ATP.lambda_362
tff(fact_9058_ATP_Olambda__363,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_it(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_363
tff(fact_9059_ATP_Olambda__364,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_il(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_364
tff(fact_9060_ATP_Olambda__365,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_in(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_365
tff(fact_9061_ATP_Olambda__366,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ip(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_366
tff(fact_9062_ATP_Olambda__367,axiom,
    ! [Uu: fun_Pr1607443439640623155t_bool,Uua: produc3843707927480180839at_nat,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(aTP_Lamm_ke(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_Pro1809015098595223084t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_367
tff(fact_9063_ATP_Olambda__368,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_kh(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_Pro2232559814647311250t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_368
tff(fact_9064_ATP_Olambda__369,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ki(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_369
tff(fact_9065_ATP_Olambda__370,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_kf(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_370
tff(fact_9066_ATP_Olambda__371,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_kg(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_371
tff(fact_9067_ATP_Olambda__372,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_kj(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_372
tff(fact_9068_ATP_Olambda__373,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_kk(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_373
tff(fact_9069_ATP_Olambda__374,axiom,
    ! [Uu: fun_Pr1607443439640623155t_bool,Uua: produc3843707927480180839at_nat,Uub: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(aTP_Lamm_jx(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_Pro1809015098595223084t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_374
tff(fact_9070_ATP_Olambda__375,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_ka(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_Pro2232559814647311250t_bool(Uu,Uub)) ) ) ).

% ATP.lambda_375
tff(fact_9071_ATP_Olambda__376,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_kb(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_376
tff(fact_9072_ATP_Olambda__377,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_jy(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_377
tff(fact_9073_ATP_Olambda__378,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_jz(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_378
tff(fact_9074_ATP_Olambda__379,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_kc(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_379
tff(fact_9075_ATP_Olambda__380,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_kd(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_380
tff(fact_9076_ATP_Olambda__381,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_ed(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_381
tff(fact_9077_ATP_Olambda__382,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_el(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_382
tff(fact_9078_ATP_Olambda__383,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_dx(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_383
tff(fact_9079_ATP_Olambda__384,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_ef(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_384
tff(fact_9080_ATP_Olambda__385,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_en(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_385
tff(fact_9081_ATP_Olambda__386,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_dz(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_386
tff(fact_9082_ATP_Olambda__387,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_eh(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_387
tff(fact_9083_ATP_Olambda__388,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_ep(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_388
tff(fact_9084_ATP_Olambda__389,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_eb(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_389
tff(fact_9085_ATP_Olambda__390,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_ej(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_390
tff(fact_9086_ATP_Olambda__391,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_dj(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_391
tff(fact_9087_ATP_Olambda__392,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_dr(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_392
tff(fact_9088_ATP_Olambda__393,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_dd(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_393
tff(fact_9089_ATP_Olambda__394,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_dl(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_394
tff(fact_9090_ATP_Olambda__395,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_dt(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_395
tff(fact_9091_ATP_Olambda__396,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_df(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_396
tff(fact_9092_ATP_Olambda__397,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_dn(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_397
tff(fact_9093_ATP_Olambda__398,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_dv(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_398
tff(fact_9094_ATP_Olambda__399,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_dh(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_399
tff(fact_9095_ATP_Olambda__400,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_dp(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_400
tff(fact_9096_ATP_Olambda__401,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zr(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).

% ATP.lambda_401
tff(fact_9097_ATP_Olambda__402,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_uy(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_402
tff(fact_9098_ATP_Olambda__403,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_aae(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu)) ) ).

% ATP.lambda_403
tff(fact_9099_ATP_Olambda__404,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aag(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu)) ) ).

% ATP.lambda_404
tff(fact_9100_ATP_Olambda__405,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_vm(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_405
tff(fact_9101_ATP_Olambda__406,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(aTP_Lamm_cg(Uu),Uua),Uub))
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_406
tff(fact_9102_ATP_Olambda__407,axiom,
    ! [Uu: set_Pr7459493094073627847at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: set_Pr4329608150637261639at_nat] :
      ( pp(aa_set4148026973699692364t_bool(aa_set6699703996738328926t_bool(aTP_Lamm_ch(Uu),Uua),Uub))
    <=> member1466754251312161552at_nat(produc9060074326276436823at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_407
tff(fact_9103_ATP_Olambda__408,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(aTP_Lamm_ci(Uu),Uua),Uub))
    <=> pp(aa_set4148026973699692364t_bool(member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uub)),Uu)) ) ).

% ATP.lambda_408
tff(fact_9104_ATP_Olambda__409,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu)) ) ).

% ATP.lambda_409
tff(fact_9105_ATP_Olambda__410,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ck(Uu),Uua),Uub))
    <=> member5262025264175285858nt_int(product_Pair_int_int(Uua,Uub),Uu) ) ).

% ATP.lambda_410
tff(fact_9106_ATP_Olambda__411,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_da(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_411
tff(fact_9107_ATP_Olambda__412,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_ye(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_412
tff(fact_9108_ATP_Olambda__413,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_413
tff(fact_9109_ATP_Olambda__414,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uk(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_414
tff(fact_9110_ATP_Olambda__415,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ul(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_415
tff(fact_9111_ATP_Olambda__416,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_vk(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_416
tff(fact_9112_ATP_Olambda__417,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_we(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_417
tff(fact_9113_ATP_Olambda__418,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_vl(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_418
tff(fact_9114_ATP_Olambda__419,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_419
tff(fact_9115_ATP_Olambda__420,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_aaz(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_420
tff(fact_9116_ATP_Olambda__421,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_yb(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_421
tff(fact_9117_ATP_Olambda__422,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_kl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub)) ).

% ATP.lambda_422
tff(fact_9118_ATP_Olambda__423,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_km(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_423
tff(fact_9119_ATP_Olambda__424,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kn(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_424
tff(fact_9120_ATP_Olambda__425,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_425
tff(fact_9121_ATP_Olambda__426,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kp(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_426
tff(fact_9122_ATP_Olambda__427,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ko(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_427
tff(fact_9123_ATP_Olambda__428,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_gv(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_428
tff(fact_9124_ATP_Olambda__429,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_429
tff(fact_9125_ATP_Olambda__430,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gu(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_430
tff(fact_9126_ATP_Olambda__431,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_sy(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_431
tff(fact_9127_ATP_Olambda__432,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gt(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_432
tff(fact_9128_ATP_Olambda__433,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_433
tff(fact_9129_ATP_Olambda__434,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_al(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2232559814647311250t_bool(Uu,Uub))
        | pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_434
tff(fact_9130_ATP_Olambda__435,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ak(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        | pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_435
tff(fact_9131_ATP_Olambda__436,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_af(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        | pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_436
tff(fact_9132_ATP_Olambda__437,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        | pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_437
tff(fact_9133_ATP_Olambda__438,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ah(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        | pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_438
tff(fact_9134_ATP_Olambda__439,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        | pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_439
tff(fact_9135_ATP_Olambda__440,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_aj(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        | pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_440
tff(fact_9136_ATP_Olambda__441,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: fun_Pr7757163693017731481t_bool,Uub: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_fun830836278467416944t_bool(aTP_Lamm_as(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2232559814647311250t_bool(Uu,Uub))
        & pp(aa_Pro2232559814647311250t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_441
tff(fact_9137_ATP_Olambda__442,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_ar(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_442
tff(fact_9138_ATP_Olambda__443,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_am(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_443
tff(fact_9139_ATP_Olambda__444,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_an(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_444
tff(fact_9140_ATP_Olambda__445,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_ao(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_445
tff(fact_9141_ATP_Olambda__446,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ap(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_446
tff(fact_9142_ATP_Olambda__447,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_aq(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_447
tff(fact_9143_ATP_Olambda__448,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_448
tff(fact_9144_ATP_Olambda__449,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tb(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_449
tff(fact_9145_ATP_Olambda__450,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_450
tff(fact_9146_ATP_Olambda__451,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aab(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_451
tff(fact_9147_ATP_Olambda__452,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_zc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_452
tff(fact_9148_ATP_Olambda__453,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_mj(Uu),Uua),Uub) = plus_p3455044024723400733d_enat(Uu,times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub),Uua)) ).

% ATP.lambda_453
tff(fact_9149_ATP_Olambda__454,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mh(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_454
tff(fact_9150_ATP_Olambda__455,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_455
tff(fact_9151_ATP_Olambda__456,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_mi(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_456
tff(fact_9152_ATP_Olambda__457,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mb(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_457
tff(fact_9153_ATP_Olambda__458,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ma(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_458
tff(fact_9154_ATP_Olambda__459,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_od(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_459
tff(fact_9155_ATP_Olambda__460,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_og(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_460
tff(fact_9156_ATP_Olambda__461,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oe(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_461
tff(fact_9157_ATP_Olambda__462,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_of(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_462
tff(fact_9158_ATP_Olambda__463,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_463
tff(fact_9159_ATP_Olambda__464,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ku(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_464
tff(fact_9160_ATP_Olambda__465,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_kx(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_465
tff(fact_9161_ATP_Olambda__466,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_kv(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_466
tff(fact_9162_ATP_Olambda__467,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_kw(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_467
tff(fact_9163_ATP_Olambda__468,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_468
tff(fact_9164_ATP_Olambda__469,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aad(Uu),Uua),Uub))
    <=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua))) ) ).

% ATP.lambda_469
tff(fact_9165_ATP_Olambda__470,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_sz(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_470
tff(fact_9166_ATP_Olambda__471,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ta(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_471
tff(fact_9167_ATP_Olambda__472,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_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_472
tff(fact_9168_ATP_Olambda__473,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_sf(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_473
tff(fact_9169_ATP_Olambda__474,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_si(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_474
tff(fact_9170_ATP_Olambda__475,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sh(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_475
tff(fact_9171_ATP_Olambda__476,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nj(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_476
tff(fact_9172_ATP_Olambda__477,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ni(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_477
tff(fact_9173_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_rt(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_478
tff(fact_9174_ATP_Olambda__479,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_kt(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_479
tff(fact_9175_ATP_Olambda__480,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aac(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_480
tff(fact_9176_ATP_Olambda__481,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_481
tff(fact_9177_ATP_Olambda__482,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ri(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_482
tff(fact_9178_ATP_Olambda__483,axiom,
    ! [Uu: fun_Pr7757163693017731481t_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fh(Uu),Uua),Uub))
    <=> pp(aa_Pro2232559814647311250t_bool(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub))) ) ).

% ATP.lambda_483
tff(fact_9179_ATP_Olambda__484,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fi(Uu),Uua),Uub) = aa_Pro4243173770044397973at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_484
tff(fact_9180_ATP_Olambda__485,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_fj(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_485
tff(fact_9181_ATP_Olambda__486,axiom,
    ! [Uu: fun_Pr248683899324676269t_bool,Uua: nat,Uub: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_fk(Uu),Uua),Uub) = aa_Pro595088089689268830t_bool(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_486
tff(fact_9182_ATP_Olambda__487,axiom,
    ! [Uu: fun_Pr803113274115125473t_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fl(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(Uu,product_Pair_int_int(Uua,Uub))) ) ).

% ATP.lambda_487
tff(fact_9183_ATP_Olambda__488,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_td(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_488
tff(fact_9184_ATP_Olambda__489,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_te(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_489
tff(fact_9185_ATP_Olambda__490,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wh(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_490
tff(fact_9186_ATP_Olambda__491,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wp(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_491
tff(fact_9187_ATP_Olambda__492,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wj(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_zero_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_492
tff(fact_9188_ATP_Olambda__493,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wn(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_493
tff(fact_9189_ATP_Olambda__494,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wl(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_494
tff(fact_9190_ATP_Olambda__495,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ws(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_495
tff(fact_9191_ATP_Olambda__496,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_496
tff(fact_9192_ATP_Olambda__497,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wt(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_497
tff(fact_9193_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wv(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_498
tff(fact_9194_ATP_Olambda__499,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wu(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,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_499
tff(fact_9195_ATP_Olambda__500,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wi(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_500
tff(fact_9196_ATP_Olambda__501,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_501
tff(fact_9197_ATP_Olambda__502,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wk(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_502
tff(fact_9198_ATP_Olambda__503,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wo(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_503
tff(fact_9199_ATP_Olambda__504,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wm(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_504
tff(fact_9200_ATP_Olambda__505,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_tr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uuc),Uua)),aa_nat_real(Uub,Uuc)),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_505
tff(fact_9201_ATP_Olambda__506,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aa_fun4268748527592579270omplex(aTP_Lamm_rw(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) ).

% ATP.lambda_506
tff(fact_9202_ATP_Olambda__507,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_lh(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_507
tff(fact_9203_ATP_Olambda__508,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_li(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_508
tff(fact_9204_ATP_Olambda__509,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_lf(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_509
tff(fact_9205_ATP_Olambda__510,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lg(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_int(Uua,Uuc)),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_510
tff(fact_9206_ATP_Olambda__511,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] : aa_real_complex(aa_fun2233226938935547116omplex(aa_fun2106233272836097472omplex(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) ).

% ATP.lambda_511
tff(fact_9207_ATP_Olambda__512,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ll(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_512
tff(fact_9208_ATP_Olambda__513,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_lm(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_513
tff(fact_9209_ATP_Olambda__514,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] : aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_lj(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_514
tff(fact_9210_ATP_Olambda__515,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] : aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_lk(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_int(Uua,Uuc)),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_515
tff(fact_9211_ATP_Olambda__516,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_516
tff(fact_9212_ATP_Olambda__517,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sa(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_517
tff(fact_9213_ATP_Olambda__518,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] : aa_int_complex(aa_fun3088484340408535532omplex(aa_fun8701580876069687872omplex(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu),aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) ).

% ATP.lambda_518
tff(fact_9214_ATP_Olambda__519,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lo(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu)),aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_519
tff(fact_9215_ATP_Olambda__520,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ln(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu)),aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_520
tff(fact_9216_ATP_Olambda__521,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mx(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mw(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_521
tff(fact_9217_ATP_Olambda__522,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mv(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mu(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_522
tff(fact_9218_ATP_Olambda__523,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ms(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_523
tff(fact_9219_ATP_Olambda__524,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ww(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_complex_complex(uminus1482373934393186551omplex,Uub)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_complex),zero_zero_complex))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_524
tff(fact_9220_ATP_Olambda__525,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wz(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat)),aa_real_real(uminus_uminus_real,Uub)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_real),zero_zero_real))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_525
tff(fact_9221_ATP_Olambda__526,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_wy(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_rat_rat(uminus_uminus_rat,Uub)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_rat),zero_zero_rat))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_526
tff(fact_9222_ATP_Olambda__527,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wx(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_int_int(uminus_uminus_int,Uub)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_int),zero_zero_int))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_527
tff(fact_9223_ATP_Olambda__528,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_int,Uub: fun_nat_fun_nat_bool,Uuc: nat] : aa_nat_int(aa_fun467087377561674726at_int(aa_fun989774750125491633at_int(aTP_Lamm_qs(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qr(Uua),Uuc)),collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bp(Uu),Uub),Uuc))) ).

% ATP.lambda_528
tff(fact_9224_ATP_Olambda__529,axiom,
    ! [Uu: set_nat,Uua: fun_na5967208032043182372ex_int,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_int(aa_fun4682754999372510822ex_int(aa_fun2078238821050448595ex_int(aTP_Lamm_qo(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_com2205869648058525157at_int(aTP_Lamm_qn(Uua),Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_br(Uu),Uub),Uuc))) ).

% ATP.lambda_529
tff(fact_9225_ATP_Olambda__530,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_int,Uub: fun_na6801867027684862029l_bool,Uuc: real] : aa_real_int(aa_fun5434943856969160678al_int(aa_fun2246985729214054869al_int(aTP_Lamm_ql(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_real_fun_nat_int(aTP_Lamm_qj(Uua),Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun6065609019945104124t_bool(aTP_Lamm_qk(Uu),Uub),Uuc))) ).

% ATP.lambda_530
tff(fact_9226_ATP_Olambda__531,axiom,
    ! [Uu: set_complex,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_pw(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_pv(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bv(Uu),Uub),Uuc))) ).

% ATP.lambda_531
tff(fact_9227_ATP_Olambda__532,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_pt(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_ps(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bm(Uu),Uub),Uuc))) ).

% ATP.lambda_532
tff(fact_9228_ATP_Olambda__533,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_qg(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hu(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bv(Uu),Uub),Uuc))) ).

% ATP.lambda_533
tff(fact_9229_ATP_Olambda__534,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_qe(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_hq(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bs(Uu),Uub),Uuc))) ).

% ATP.lambda_534
tff(fact_9230_ATP_Olambda__535,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_qc(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_hm(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bm(Uu),Uub),Uuc))) ).

% ATP.lambda_535
tff(fact_9231_ATP_Olambda__536,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: int] : aa_int_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_qa(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_hh(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_hi(Uu),Uub),Uuc))) ).

% ATP.lambda_536
tff(fact_9232_ATP_Olambda__537,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: int] : aa_int_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_py(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_gz(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bn(Uu),Uub),Uuc))) ).

% ATP.lambda_537
tff(fact_9233_ATP_Olambda__538,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_real(aa_fun555827300007793674t_real(aa_fun8353879187736340217t_real(aTP_Lamm_ik(Uu),Uua),Uub),Uuc) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_ij(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bm(Uu),Uub),Uuc))) ).

% ATP.lambda_538
tff(fact_9234_ATP_Olambda__539,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: complex] : aa_complex_complex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_ih(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_ig(Uua),Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_bu(Uu),Uub),Uuc))) ).

% ATP.lambda_539
tff(fact_9235_ATP_Olambda__540,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_complex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_id(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ic(Uua),Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_br(Uu),Uub),Uuc))) ).

% ATP.lambda_540
tff(fact_9236_ATP_Olambda__541,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_hz(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_hy(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bo(Uu),Uub),Uuc))) ).

% ATP.lambda_541
tff(fact_9237_ATP_Olambda__542,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_hv(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hu(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bv(Uu),Uub),Uuc))) ).

% ATP.lambda_542
tff(fact_9238_ATP_Olambda__543,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_hr(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_hq(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bs(Uu),Uub),Uuc))) ).

% ATP.lambda_543
tff(fact_9239_ATP_Olambda__544,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_hn(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_hm(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bm(Uu),Uub),Uuc))) ).

% ATP.lambda_544
tff(fact_9240_ATP_Olambda__545,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: int] : aa_int_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_hj(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_hh(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_hi(Uu),Uub),Uuc))) ).

% ATP.lambda_545
tff(fact_9241_ATP_Olambda__546,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_bool,Uuc: int] : aa_int_int(aa_fun4504283173926863078nt_int(aa_fun5512849559334105941nt_int(aTP_Lamm_he(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hd(Uua),Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_546
tff(fact_9242_ATP_Olambda__547,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: int] : aa_int_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_ha(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_gz(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bn(Uu),Uub),Uuc))) ).

% ATP.lambda_547
tff(fact_9243_ATP_Olambda__548,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_wg(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_wf(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_548
tff(fact_9244_ATP_Olambda__549,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_vv(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_vu(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_549
tff(fact_9245_ATP_Olambda__550,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_vt(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_vs(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_550
tff(fact_9246_ATP_Olambda__551,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_vr(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_vq(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_551
tff(fact_9247_ATP_Olambda__552,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_vp(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_vo(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_552
tff(fact_9248_ATP_Olambda__553,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_zn(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_553
tff(fact_9249_ATP_Olambda__554,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_zl(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_554
tff(fact_9250_ATP_Olambda__555,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_zk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_555
tff(fact_9251_ATP_Olambda__556,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_zj(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_556
tff(fact_9252_ATP_Olambda__557,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(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_557
tff(fact_9253_ATP_Olambda__558,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_558
tff(fact_9254_ATP_Olambda__559,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_np(Uu),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_559
tff(fact_9255_ATP_Olambda__560,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nr(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_560
tff(fact_9256_ATP_Olambda__561,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yk(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_561
tff(fact_9257_ATP_Olambda__562,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ms(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_562
tff(fact_9258_ATP_Olambda__563,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_563
tff(fact_9259_ATP_Olambda__564,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uua),Uu)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_564
tff(fact_9260_ATP_Olambda__565,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wa(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_565
tff(fact_9261_ATP_Olambda__566,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wd(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_566
tff(fact_9262_ATP_Olambda__567,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_wb(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_567
tff(fact_9263_ATP_Olambda__568,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wc(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_568
tff(fact_9264_ATP_Olambda__569,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vw(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_569
tff(fact_9265_ATP_Olambda__570,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_vz(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_570
tff(fact_9266_ATP_Olambda__571,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_vx(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_571
tff(fact_9267_ATP_Olambda__572,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vn(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_572
tff(fact_9268_ATP_Olambda__573,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vy(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_573
tff(fact_9269_ATP_Olambda__574,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nv(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_574
tff(fact_9270_ATP_Olambda__575,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ny(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_575
tff(fact_9271_ATP_Olambda__576,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nw(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_576
tff(fact_9272_ATP_Olambda__577,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nx(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_577
tff(fact_9273_ATP_Olambda__578,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_hw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_578
tff(fact_9274_ATP_Olambda__579,axiom,
    ! [Uu: set_complex,Uua: fun_na5941084329111710283x_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_ia(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_579
tff(fact_9275_ATP_Olambda__580,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_ie(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_580
tff(fact_9276_ATP_Olambda__581,axiom,
    ! [Uu: set_real,Uua: fun_na6801867027684862029l_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3745989723528014332l_bool(aTP_Lamm_qh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_nat_fun_real_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_581
tff(fact_9277_ATP_Olambda__582,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_hs(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_582
tff(fact_9278_ATP_Olambda__583,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_hk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_583
tff(fact_9279_ATP_Olambda__584,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_qp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_584
tff(fact_9280_ATP_Olambda__585,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ho(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_585
tff(fact_9281_ATP_Olambda__586,axiom,
    ! [Uu: set_int,Uua: fun_co2381122952212249071t_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_586
tff(fact_9282_ATP_Olambda__587,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real,Uuc: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_gx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_587
tff(fact_9283_ATP_Olambda__588,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun7617171277304647676t_bool(aTP_Lamm_hb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_588
tff(fact_9284_ATP_Olambda__589,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_bv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_589
tff(fact_9285_ATP_Olambda__590,axiom,
    ! [Uu: set_complex,Uua: fun_co2381122952212249071t_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_hi(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_int_bool(aa_com2211550117870696024t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_590
tff(fact_9286_ATP_Olambda__591,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_591
tff(fact_9287_ATP_Olambda__592,axiom,
    ! [Uu: set_real,Uua: fun_re9121486324101951821t_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_592
tff(fact_9288_ATP_Olambda__593,axiom,
    ! [Uu: set_real,Uua: fun_re8661203372455305201t_bool,Uub: int,Uuc: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_bn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_593
tff(fact_9289_ATP_Olambda__594,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_br(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_594
tff(fact_9290_ATP_Olambda__595,axiom,
    ! [Uu: set_nat,Uua: fun_na6801867027684862029l_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun6065609019945104124t_bool(aTP_Lamm_qk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_real_bool(aa_nat_fun_real_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_595
tff(fact_9291_ATP_Olambda__596,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_596
tff(fact_9292_ATP_Olambda__597,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_597
tff(fact_9293_ATP_Olambda__598,axiom,
    ! [Uu: set_int,Uua: fun_in5750378413879446255x_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_bu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_598
tff(fact_9294_ATP_Olambda__599,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bs(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_599
tff(fact_9295_ATP_Olambda__600,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_bool,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_bt(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_int_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_600
tff(fact_9296_ATP_Olambda__601,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_no(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_601
tff(fact_9297_ATP_Olambda__602,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_602
tff(fact_9298_ATP_Olambda__603,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nq(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_603
tff(fact_9299_ATP_Olambda__604,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ns(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_604
tff(fact_9300_ATP_Olambda__605,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nz(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_605
tff(fact_9301_ATP_Olambda__606,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_oc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_606
tff(fact_9302_ATP_Olambda__607,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_oa(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_607
tff(fact_9303_ATP_Olambda__608,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ob(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_608
tff(fact_9304_ATP_Olambda__609,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vh(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_609
tff(fact_9305_ATP_Olambda__610,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yp(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_610
tff(fact_9306_ATP_Olambda__611,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_yn(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_611
tff(fact_9307_ATP_Olambda__612,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yr(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_612
tff(fact_9308_ATP_Olambda__613,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_yt(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_613
tff(fact_9309_ATP_Olambda__614,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_dk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_614
tff(fact_9310_ATP_Olambda__615,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_ds(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_615
tff(fact_9311_ATP_Olambda__616,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_de(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_616
tff(fact_9312_ATP_Olambda__617,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_dm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_617
tff(fact_9313_ATP_Olambda__618,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_du(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_618
tff(fact_9314_ATP_Olambda__619,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_dg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_619
tff(fact_9315_ATP_Olambda__620,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_do(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_620
tff(fact_9316_ATP_Olambda__621,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_dw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_621
tff(fact_9317_ATP_Olambda__622,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_di(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_622
tff(fact_9318_ATP_Olambda__623,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_dq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_623
tff(fact_9319_ATP_Olambda__624,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ee(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_624
tff(fact_9320_ATP_Olambda__625,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_em(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_625
tff(fact_9321_ATP_Olambda__626,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_dy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_626
tff(fact_9322_ATP_Olambda__627,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_eg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_627
tff(fact_9323_ATP_Olambda__628,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_eo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_628
tff(fact_9324_ATP_Olambda__629,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_ea(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_629
tff(fact_9325_ATP_Olambda__630,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_ei(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_630
tff(fact_9326_ATP_Olambda__631,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_eq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_631
tff(fact_9327_ATP_Olambda__632,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ec(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_632
tff(fact_9328_ATP_Olambda__633,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ek(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_633
tff(fact_9329_ATP_Olambda__634,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_vo(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_634
tff(fact_9330_ATP_Olambda__635,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_vu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_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_635
tff(fact_9331_ATP_Olambda__636,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_vq(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_636
tff(fact_9332_ATP_Olambda__637,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_wf(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_637
tff(fact_9333_ATP_Olambda__638,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_vs(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_638
tff(fact_9334_ATP_Olambda__639,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_int,Uub: fun_nat_fun_nat_bool,Uuc: nat] : aa_nat_int(aa_fun467087377561674726at_int(aa_fun989774750125491633at_int(aTP_Lamm_qq(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uua,Uuc)),collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_qp(Uu),Uub),Uuc))) ).

% ATP.lambda_639
tff(fact_9335_ATP_Olambda__640,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_int,Uub: fun_na6801867027684862029l_bool,Uuc: nat] : aa_nat_int(aa_fun3886942400138156042at_int(aa_fun699103053049729529at_int(aTP_Lamm_qi(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(Uua,Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3745989723528014332l_bool(aTP_Lamm_qh(Uu),Uub),Uuc))) ).

% ATP.lambda_640
tff(fact_9336_ATP_Olambda__641,axiom,
    ! [Uu: set_complex,Uua: fun_na5967208032043182372ex_int,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_int(aa_fun3278734219653515656at_int(aa_fun9069282223758319861at_int(aTP_Lamm_qm(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_ia(Uu),Uub),Uuc))) ).

% ATP.lambda_641
tff(fact_9337_ATP_Olambda__642,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_qf(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_hs(Uu),Uub),Uuc))) ).

% ATP.lambda_642
tff(fact_9338_ATP_Olambda__643,axiom,
    ! [Uu: set_nat,Uua: fun_co7370231399432144932at_int,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_pu(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_hs(Uu),Uub),Uuc))) ).

% ATP.lambda_643
tff(fact_9339_ATP_Olambda__644,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_qd(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ho(Uu),Uub),Uuc))) ).

% ATP.lambda_644
tff(fact_9340_ATP_Olambda__645,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_qb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_hk(Uu),Uub),Uuc))) ).

% ATP.lambda_645
tff(fact_9341_ATP_Olambda__646,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_int,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_pr(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_hk(Uu),Uub),Uuc))) ).

% ATP.lambda_646
tff(fact_9342_ATP_Olambda__647,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: complex] : aa_complex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_pz(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hf(Uu),Uub),Uuc))) ).

% ATP.lambda_647
tff(fact_9343_ATP_Olambda__648,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: real] : aa_real_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_px(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_gx(Uu),Uub),Uuc))) ).

% ATP.lambda_648
tff(fact_9344_ATP_Olambda__649,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: int] : aa_int_complex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_if(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_ie(Uu),Uub),Uuc))) ).

% ATP.lambda_649
tff(fact_9345_ATP_Olambda__650,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_complex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_ib(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_ia(Uu),Uub),Uuc))) ).

% ATP.lambda_650
tff(fact_9346_ATP_Olambda__651,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_hx(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_hw(Uu),Uub),Uuc))) ).

% ATP.lambda_651
tff(fact_9347_ATP_Olambda__652,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_ht(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_hs(Uu),Uub),Uuc))) ).

% ATP.lambda_652
tff(fact_9348_ATP_Olambda__653,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_hp(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_ho(Uu),Uub),Uuc))) ).

% ATP.lambda_653
tff(fact_9349_ATP_Olambda__654,axiom,
    ! [Uu: set_nat,Uua: fun_re4238438383274954274t_real,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_real(aa_fun7981600115099051494l_real(aa_fun2847282595593119957l_real(aTP_Lamm_ii(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_hk(Uu),Uub),Uuc))) ).

% ATP.lambda_654
tff(fact_9350_ATP_Olambda__655,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_hl(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_hk(Uu),Uub),Uuc))) ).

% ATP.lambda_655
tff(fact_9351_ATP_Olambda__656,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co2381122952212249071t_bool,Uuc: complex] : aa_complex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_hg(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hf(Uu),Uub),Uuc))) ).

% ATP.lambda_656
tff(fact_9352_ATP_Olambda__657,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_bool,Uuc: nat] : aa_nat_int(aa_fun3504929462025808522at_int(aa_fun3325927984641084153at_int(aTP_Lamm_hc(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_bool(aa_fun7617171277304647676t_bool(aTP_Lamm_hb(Uu),Uub),Uuc))) ).

% ATP.lambda_657
tff(fact_9353_ATP_Olambda__658,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_re8661203372455305201t_bool,Uuc: real] : aa_real_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_gy(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_gx(Uu),Uub),Uuc))) ).

% ATP.lambda_658
tff(fact_9354_ATP_Olambda__659,axiom,
    ! [Uu: bool,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_fp(Uu),Uua),Uub),Uuc))
    <=> ( pp(Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_659
tff(fact_9355_ATP_Olambda__660,axiom,
    ! [Uu: bool,Uua: fun_int_fun_int_bool,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_fq(Uu),Uua),Uub),Uuc))
    <=> ( pp(Uu)
        & pp(aa_int_bool(aa_int_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_660
tff(fact_9356_ATP_Olambda__661,axiom,
    ! [Uu: fun_bool_bool,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_ex(Uu),Uua),Uub),Uuc))
    <=> pp(aa_bool_bool(Uu,aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Uuc))) ) ).

% ATP.lambda_661
tff(fact_9357_ATP_Olambda__662,axiom,
    ! [Uu: fun_bool_bool,Uua: fun_int_fun_int_bool,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_fb(Uu),Uua),Uub),Uuc))
    <=> pp(aa_bool_bool(Uu,aa_int_bool(aa_int_fun_int_bool(Uua,Uub),Uuc))) ) ).

% ATP.lambda_662
tff(fact_9358_ATP_Olambda__663,axiom,
    ! [Uu: fun_bool_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun8134142976681035223at_nat(aTP_Lamm_ey(Uu),Uua),Uub),Uuc) = aa_bool_nat(Uu,aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Uuc)) ).

% ATP.lambda_663
tff(fact_9359_ATP_Olambda__664,axiom,
    ! [Uu: fun_bo3682073979402284333at_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aa_fun8974016276556183434at_nat(aTP_Lamm_fg(Uu),Uua),Uub),Uuc) = aa_boo5791090712870310366at_nat(Uu,aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Uuc)) ).

% ATP.lambda_664
tff(fact_9360_ATP_Olambda__665,axiom,
    ! [Uu: fun_bo6148171183121378627t_bool,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aa_fun3328218794057202080t_bool(aTP_Lamm_fc(Uu),Uua),Uub),Uuc) = aa_boo3692867584632636148t_bool(Uu,aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uub),Uuc)) ).

% ATP.lambda_665
tff(fact_9361_ATP_Olambda__666,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun1421985582829848961t_bool(aTP_Lamm_ez(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc))) ) ).

% ATP.lambda_666
tff(fact_9362_ATP_Olambda__667,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_fa(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_667
tff(fact_9363_ATP_Olambda__668,axiom,
    ! [Uu: fun_na4827981024643048934t_bool,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aa_fun9209259830566968027t_bool(aTP_Lamm_fd(Uu),Uua),Uub),Uuc) = aa_nat4401263874371387303t_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_668
tff(fact_9364_ATP_Olambda__669,axiom,
    ! [Uu: fun_fu5641360819561228055l_bool,Uua: fun_na4786964484885125043t_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun7513403936370313208t_bool(aTP_Lamm_fe(Uu),Uua),Uub),Uuc))
    <=> pp(aa_fun8663595511526100224l_bool(Uu,aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(Uua,Uub),Uuc))) ) ).

% ATP.lambda_669
tff(fact_9365_ATP_Olambda__670,axiom,
    ! [Uu: fun_fu4450724535874491238ol_nat,Uua: fun_na4786964484885125043t_bool,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun3048062428331337179at_nat(aTP_Lamm_ff(Uu),Uua),Uub),Uuc) = aa_fun3448868659104709799ol_nat(Uu,aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(Uua,Uub),Uuc)) ).

% ATP.lambda_670
tff(fact_9366_ATP_Olambda__671,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_kz(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_671
tff(fact_9367_ATP_Olambda__672,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ky(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_672
tff(fact_9368_ATP_Olambda__673,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ru(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_673
tff(fact_9369_ATP_Olambda__674,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_xl(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_xk(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_674
tff(fact_9370_ATP_Olambda__675,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_xj(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_xi(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_675
tff(fact_9371_ATP_Olambda__676,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_xh(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_xg(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_676
tff(fact_9372_ATP_Olambda__677,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_xf(Uu),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_xe(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_677
tff(fact_9373_ATP_Olambda__678,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_zm(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_zl(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_678
tff(fact_9374_ATP_Olambda__679,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_xe(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_679
tff(fact_9375_ATP_Olambda__680,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_xk(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_680
tff(fact_9376_ATP_Olambda__681,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: rat,Uuc: nat,Uud: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_xg(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(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_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_681
tff(fact_9377_ATP_Olambda__682,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_xi(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_682
tff(fact_9378_ATP_Olambda__683,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_aaj(Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))
           => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_13))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I4)) ) )
        & ( ( Uuc = Uud )
         => ! [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),Uud))
            & ! [X5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),X5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X5))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_683
tff(fact_9379_ATP_Olambda__684,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gr,Uu),Uua))
    <=> $true ) ).

% ATP.lambda_684
tff(fact_9380_ATP_Olambda__685,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_gs,Uu),Uua))
    <=> $true ) ).

% ATP.lambda_685
tff(fact_9381_ATP_Olambda__686,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cb,Uu) = Uu ).

% ATP.lambda_686
tff(fact_9382_ATP_Olambda__687,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_cc,Uu) = Uu ).

% ATP.lambda_687
tff(fact_9383_ATP_Olambda__688,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cd,Uu) = Uu ).

% ATP.lambda_688
tff(fact_9384_ATP_Olambda__689,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ce,Uu) = Uu ).

% ATP.lambda_689
tff(fact_9385_ATP_Olambda__690,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cf,Uu) = Uu ).

% ATP.lambda_690
tff(fact_9386_ATP_Olambda__691,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_bw,Uu) = zero_zero_complex ).

% ATP.lambda_691
tff(fact_9387_ATP_Olambda__692,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_bx,Uu) = zero_zero_real ).

% ATP.lambda_692
tff(fact_9388_ATP_Olambda__693,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_by,Uu) = zero_zero_rat ).

% ATP.lambda_693
tff(fact_9389_ATP_Olambda__694,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_ok,Uu) = zero_zero_complex ).

% ATP.lambda_694
tff(fact_9390_ATP_Olambda__695,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_fr,Uu) = zero_zero_real ).

% ATP.lambda_695
tff(fact_9391_ATP_Olambda__696,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bz,Uu) = zero_zero_nat ).

% ATP.lambda_696
tff(fact_9392_ATP_Olambda__697,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ol,Uu) = zero_zero_int ).

% ATP.lambda_697
tff(fact_9393_ATP_Olambda__698,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ca,Uu) = zero_zero_int ).

% ATP.lambda_698
tff(fact_9394_ATP_Olambda__699,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_bi,Uu))
    <=> $false ) ).

% ATP.lambda_699
tff(fact_9395_ATP_Olambda__700,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_bj,Uu))
    <=> $false ) ).

% ATP.lambda_700
tff(fact_9396_ATP_Olambda__701,axiom,
    ! [Uu: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_bg,Uu))
    <=> $false ) ).

% ATP.lambda_701
tff(fact_9397_ATP_Olambda__702,axiom,
    ! [Uu: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_bh,Uu))
    <=> $false ) ).

% ATP.lambda_702
tff(fact_9398_ATP_Olambda__703,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_bk,Uu))
    <=> $false ) ).

% ATP.lambda_703
tff(fact_9399_ATP_Olambda__704,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_bl,Uu))
    <=> $false ) ).

% ATP.lambda_704
tff(fact_9400_ATP_Olambda__705,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_yf,Uu))
    <=> $true ) ).

% ATP.lambda_705

% Helper facts (60)
tff(help_fNot_2_1_U,axiom,
    ! [P2: bool] :
      ( pp(P2)
      | pp(aa_bool_bool(fNot,P2)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P2: bool] :
      ( ~ pp(aa_bool_bool(fNot,P2))
      | ~ pp(P2) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P2),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P2),Q))
      | pp(P2) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(P2)
      | ~ pp(Q)
      | pp(aa_bool_bool(fconj(P2),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fdisj(P2),Q))
      | pp(P2)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P2: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fdisj(P2),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(P2)
      | pp(aa_bool_bool(fdisj(P2),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P2),Q))
      | ~ pp(P2)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P2: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P2),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(P2)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P2),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fTrue,X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P2: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P2,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P2)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(fequal_bool(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(fequal_bool(X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P2: fun_real_bool] : aa_real_bool(P2,fChoice_real(P2)) = aa_fun3352386287069783311l_bool(fEx_real,P2) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( X != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : if_Extended_enat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : if_Extended_enat(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fTrue,X,Y) = X ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P2: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P2))
      | pp(aa_VEBT_VEBT_bool(P2,X)) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X != Y )
      | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( X != Y )
      | pp(aa_list_nat_bool(fequal_list_nat(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ~ pp(aa_list_nat_bool(fequal_list_nat(X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y) = X ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P2: bool] :
      ( ( P2 = fTrue )
      | ( P2 = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X != Y )
      | pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ~ pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(fequal9188845342289841878at_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P2: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P2,Q),R) = aa_bool_bool(P2,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P2: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P2,Q),R) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P2,R),aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P2: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P2,Q),R) = aa_fun_nat_bool_bool(P2,aa_VEB4832260192069861672t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P2: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P2,Q),R) = aa_boo8970175072276269406l_bool(P2,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P2: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P2,Q),R) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P2,R),Q) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( X != Y )
      | pp(aa_Pro1809015098595223084t_bool(fequal7949547063978738364at_nat(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ pp(aa_Pro1809015098595223084t_bool(fequal7949547063978738364at_nat(X),Y))
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),ya))
    | ( xa = ya ) ) ).

%------------------------------------------------------------------------------