%------------------------------------------------------------------------------
% File     : ITP232_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 00398_025158
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0067_VEBT_InsertCorrectness_00398_025158 [Des22]

% Status   : Theorem
% Rating   : 0.33 v8.2.0, 0.30 v8.1.0
% Syntax   : Number of formulae    : 12949 (3448 unt;2550 typ;   0 def)
%            Number of atoms       : 27328 (9314 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 19792 (2863   ~; 549   |;1932   &)
%                                         (2750 <=>;11698  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   15 (   2 avg)
%            Number of types       :  489 ( 488 usr)
%            Number of type conns  : 2414 (1664   >; 750   *;   0   +;   0  <<)
%            Number of predicates  :   76 (  73 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1989 (1989 usr; 398 con; 0-3 aty)
%            Number of variables   : 26053 (25337   !; 716   ?;26053   :)
% 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 20:55:18.097
%------------------------------------------------------------------------------
% Could-be-implicit typings (488)
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tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Rat__Orat_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Num__Onum_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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tff(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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tff(ty_n_t__VEBT____Definitions__OVEBT,type,
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tff(ty_n_t__Set__Oset_It__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__Extended____Nat__Oenat,type,
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tff(ty_n_t__Complex__Ocomplex,type,
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tff(ty_n_t__Nat__Onat,type,
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tff(sy_c_ATP_058Lamm__aab____,type,
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tff(sy_c_ATP_058Lamm__aac____,type,
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tff(sy_c_ATP_058Lamm__aad____,type,
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tff(sy_c_ATP_058Lamm__aae____,type,
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tff(sy_c_ATP_058Lamm__aaf____,type,
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tff(sy_c_ATP_058Lamm__aag____,type,
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tff(sy_c_ATP_058Lamm__aah____,type,
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tff(sy_c_ATP_058Lamm__aaj____,type,
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tff(sy_c_ATP_058Lamm__aak____,type,
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tff(sy_c_ATP_058Lamm__aam____,type,
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tff(sy_c_ATP_058Lamm__aan____,type,
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tff(sy_c_ATP_058Lamm__aao____,type,
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tff(sy_c_ATP_058Lamm__aaq____,type,
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tff(sy_c_ATP_058Lamm__aar____,type,
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tff(sy_c_ATP_058Lamm__aas____,type,
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tff(sy_c_ATP_058Lamm__aat____,type,
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tff(sy_c_ATP_058Lamm__aau____,type,
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tff(sy_c_ATP_058Lamm__aav____,type,
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tff(sy_c_ATP_058Lamm__aaw____,type,
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tff(sy_c_ATP_058Lamm__aax____,type,
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tff(sy_c_ATP_058Lamm__aay____,type,
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tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_na1668100129766719005at_nat ).

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_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: fun_int_nat ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_real_real ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: real > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_real_bool ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_fun_nat_rat_bool ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: rat > fun_int_fun_int_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_real_bool > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_se707272970763642173t_bool ).

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

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_VEBT_VEBT > fun_fu5250730037759307465T_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_real_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_int_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_VEBT_VEBT > fun_fu2721280555464079775T_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_VEBT_VEBT > fun_fu7178057608955062431T_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_VEBT_VEBT > fun_fu748158482767045919T_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_real_real ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_int_int ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_real_real ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_int_int ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_nat_bool > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_complex > fun_na376774557589183419x_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: nat > fun_co1378794544173401837x_bool ).

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

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_VEBT_VEBT > fun_fu8277615433164260398T_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_VEBT_VEBT > fun_fu1247336914219032636T_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_VEBT_VEBT > fun_fu8277615433164260398T_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_VEBT_VEBT > fun_fu1247336914219032636T_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_VEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_VEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_list_nat_bool > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_set_nat_bool > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_int_bool > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_complex > fun_nat_complex ).

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

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

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: vEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_int > fun_fu4313891142084863779t_bool ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_int > fun_fu3599772511451410272BT_int ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_VE294566951569886236nt_int > fun_in1463024724634152352BT_int ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_VEBT_VEBT > fun_fu3522331824740900180nt_int ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_int > fun_fu3139726878529933315t_bool ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: set_int > fun_fu3078129256884107011t_bool ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_int > fun_fu2946920718754786320at_int ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: set_nat > fun_fu6165624312919156332nt_int ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: set_int > fun_fu4844541694308389507t_bool ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: set_complex > fun_fu8510821397121033091x_bool ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_nat > fun_fu6002750694894812635t_bool ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_nat > fun_fu6772184894847113056BT_nat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_VE597728521808591204at_nat > fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_VEBT_VEBT > fun_fu4406026827936152384at_nat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_complex > fun_fu5159174019657214751x_bool ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_complex > fun_fu8973321760839499616omplex ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_VE2006399705683203104omplex > fun_co3541073130412562976omplex ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: set_VEBT_VEBT > fun_fu3138954370166276442omplex ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_bool > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: vEBT_VEBT > fun_fu7059288016784409353BT_int ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_VE5569070535936029429omplex > fun_fu6701721618212012456omplex ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_VEBT_VEBT_rat > fun_fu8873448384659826898BT_rat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_VEBT_VEBT_nat > fun_fu5238640373165125763BT_nat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_nat > fun_fu7002938529518989751at_nat ).

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

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_nat_rat > fun_nat_rat ).

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

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

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_int > fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_complex_nat > fun_complex_complex ).

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

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

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

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_int_int ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_real_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_nat > fun_nat_fun_nat_nat ).

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

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

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

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

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: complex > fun_co6883864892131739507omplex ).

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

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: rat > fun_ra5513060666784058567at_rat ).

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

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: int > fun_in6793278964266839919at_int ).

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

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_real > fun_fu6899955116716589935t_real ).

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

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

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_nat_real ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: real > fun_na6724076748692969250l_real ).

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

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

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: complex > fun_nat_complex ).

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: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: set_nat > fun_fu3446192182781468256BT_int ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_VE7331017413731640256at_int > fun_na3717937882823726332BT_int ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: set_VEBT_VEBT > fun_fu9054112635728933880at_int ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: set_nat > fun_fu6487107696295728876al_int ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: set_real > fun_fu1164691120062057232at_int ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: set_nat > fun_fu1666499901826068458ex_int ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: set_complex > fun_fu3438847080485518604at_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: set_int > fun_fu3599772511451410272BT_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: set_VEBT_VEBT > fun_fu3522331824740900180nt_int ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: set_int > fun_fu4310402290150085684al_int ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: set_real > fun_fu5129330665597888180nt_int ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: set_int > fun_fu2836861327644265266ex_int ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: set_complex > fun_fu6400036368084001456nt_int ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: set_nat > fun_fu6772184894847113056BT_nat ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: set_VEBT_VEBT > fun_fu4406026827936152384at_nat ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_int > fun_nat_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_VE5569070535936029429omplex > fun_fu6701721618212012456omplex ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_complex_complex > fun_fu248857536038387758omplex ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_real > fun_fu9152424228179946668t_real ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_complex_real > fun_fu1911030855436031788x_real ).

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

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

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_complex_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_real_real > fun_real_bool ).

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

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_real_nat > fun_real_bool ).

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

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: nat > fun_nat_fun_nat_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_rat > fun_rat_fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: nat > fun_fu5304001865264412401at_rat ).

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

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_complex > fun_co8378349014452122532t_real ).

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

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_real > fun_na5225841814499866095t_real ).

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

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_fun_nat_nat > fun_nat_nat ).

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

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

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

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

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_nat_fun_nat_nat > fun_nat_nat ).

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_VEBT_VEBT_real > fun_VEBT_VEBT_real ).

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

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_real_real > fun_real_real ).

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

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_complex_real > fun_complex_real ).

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

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

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: nat > fun_fu960360787657196261at_int ).

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_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: nat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_fun_nat_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: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: nat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_VE9219535225822561060nteger > fun_fu3480337415675744067nteger ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_VEBT_VEBT_real > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_VEBT_VEBT_rat > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: nat > fun_re6135230694098607860t_real ).

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

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: nat > fun_re6135230694098607860t_real ).

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

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: real > fun_fu6899955116716589935t_real ).

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_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: real > fun_nat_real ).

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__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
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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__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
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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,
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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__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
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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,
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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_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc7417142470700560111r_bool ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

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 * product_prod_int_int ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
    unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

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__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: ( num * num ) > produc8923325533196201883nteger ).

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__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: ( num * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

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__Code____Numeral__Ointeger,type,
    comm_s8582702949713902594nteger: code_integer > fun_nat_Code_integer ).

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

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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    lattic4729654577720512673ex_rat: ( fun_complex_rat * set_complex ) > complex ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    lattic8794016678065449205x_real: ( fun_complex_real * set_complex ) > complex ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Num__Onum,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__VEBT____Definitions__OVEBT_001t__Num__Onum,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
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tff(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: ( list_int * list_int ) > list_int ).

tff(sy_c_List_Oappend_001t__Nat__Onat,type,
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tff(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Ocount__list_001t__HOL__Obool,type,
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tff(sy_c_List_Ocount__list_001t__Int__Oint,type,
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tff(sy_c_List_Ocount__list_001t__Nat__Onat,type,
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tff(sy_c_List_Ocount__list_001t__Real__Oreal,type,
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tff(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Odistinct_001t__Int__Oint,type,
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tff(sy_c_List_Odistinct_001t__Nat__Onat,type,
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tff(sy_c_List_Odrop_001t__Nat__Onat,type,
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tff(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

tff(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001t__HOL__Obool,type,
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tff(sy_c_List_Olist__update_001t__Int__Oint,type,
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tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
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tff(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__HOL__Obool,type,
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tff(sy_c_List_Onth_001t__Nat__Onat,type,
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tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,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_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,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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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,
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tff(sy_c_List_Oupto,type,
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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,
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tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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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,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
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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__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_Orderings_Oord__class_Omax_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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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,
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tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__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,
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tff(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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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__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__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__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__Real__Oreal_001t__Real__Oreal_001t__HOL__Obool,type,
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tff(sy_c_Rat_OFract,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,
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tff(sy_c_Rat_Oquotient__of,type,
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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_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_Relation_OField_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__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__Code____Numeral__Ointeger,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__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
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tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_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__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_Set_OCollect_001t__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,
    collect_set_int: fun_set_int_bool > set_set_int ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: fun_set_nat_bool > set_set_nat ).

tff(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * set_int ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( fun_nat_int * set_nat ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( fun_nat_set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    image_5971271580939081552omplex: ( fun_re3844104906058862192omplex * set_real ) > set_fi4554929511873752355omplex ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    image_2178119161166701260l_real: ( fun_re5512370375968561900l_real * set_real ) > set_fi7789364187291644575l_real ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( fun_real_real * set_real ) > set_real ).

tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
    insert_complex: ( complex * set_complex ) > set_complex ).

tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

tff(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: ( list_nat * set_list_nat ) > set_list_nat ).

tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: ( nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Num__Onum,type,
    insert_num: ( num * set_num ) > set_num ).

tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: ( rat * set_rat ) > set_rat ).

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,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat * nat ) > fun_complex_complex ).

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

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( fun_na6724076748692969250l_real * nat ) > fun_na6724076748692969250l_real ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: 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: ( int * int ) > set_int ).

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: ( int * int ) > set_int ).

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

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__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__Int__Oint,type,
    topolo4899668324122417113eq_int: fun_nat_int > $o ).

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

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

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

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
    topolo3100542954746470799et_int: fun_nat_set_int > $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__Complex__Ocomplex,type,
    cot_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

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

tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_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 ) > fun_VE3165209603783583043T_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: fun_VE3763956218935105199t_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: fun_VE3763956218935105199t_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: fun_VE3763956218935105199t_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: fun_VE3763956218935105199t_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,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: fun_li8598321315282354825t_bool > fun_list_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_bool > fun_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( fun_Pr3099820731930151597t_bool * product_prod_int_int ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr817965127378462947t_bool * produc9072475918466114483BT_nat ) > $o ).

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
    aa_Code_integer_bool: ( fun_Co8822979462182099027r_bool * code_integer ) > bool ).

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod6977894654539884864nteger: ( fun_Co6597186368970283863nteger * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    aa_Cod2825494507867081354r_bool: ( fun_Co7333321365496032713r_bool * code_integer ) > produc7417142470700560111r_bool ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    aa_Cod7250484176784686200r_bool: ( fun_Co8310303586393106575r_bool * code_integer ) > fun_Co8822979462182099027r_bool ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    aa_Cod1242632294114111009er_int: ( fun_Co1092983284983208416er_int * code_integer ) > fun_Code_integer_int ).

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tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J,type,
    aa_fun978093584635342820t_bool: ( fun_fu4313891142084863779t_bool * fun_VE3303673267288458579t_bool ) > fun_VE3303673267288458579t_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun3638225402958239493nt_int: ( fun_fu3522331824740900180nt_int * fun_VE294566951569886236nt_int ) > fun_fu1076773466057892517nt_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    aa_fun627869447348354137BT_int: ( fun_fu3599772511451410272BT_int * fun_VE294566951569886236nt_int ) > fun_fu8394515377879832761BT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun1384843155787029862at_int: ( fun_fu5075381843755512741at_int * fun_VE3763956218935105199t_bool ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun5562694175296226570at_nat: ( fun_fu7565464988687239497at_nat * fun_VE3763956218935105199t_bool ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
    aa_fun4018725027000996448T_bool: ( fun_fu2721280555464079775T_bool * fun_VE3763956218935105199t_bool ) > fun_na2351039247863542899T_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    aa_fun6141810409722670662BT_int: ( fun_fu697195148766750301BT_int * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    aa_fun1096289392377091562BT_nat: ( fun_fu3187278293698477057BT_nat * fun_VE3763956218935105199t_bool ) > fun_VEBT_VEBT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
    aa_fun5431641998072558748t_bool: ( fun_fu6002750694894812635t_bool * fun_VE3763956218935105199t_bool ) > fun_VE3763956218935105199t_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    aa_fun6066863168309489065at_int: ( fun_fu9054112635728933880at_int * fun_VE7331017413731640256at_int ) > fun_fu5075381843755512741at_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Int__Oint_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    aa_fun6474440919679335769BT_int: ( fun_fu3446192182781468256BT_int * fun_VE7331017413731640256at_int ) > fun_fu697195148766750301BT_int ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun8318684123515291377at_nat: ( fun_fu4406026827936152384at_nat * fun_VE597728521808591204at_nat ) > fun_fu7565464988687239497at_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    aa_fun7476968357888788569BT_nat: ( fun_fu6772184894847113056BT_nat * fun_VE597728521808591204at_nat ) > fun_fu3187278293698477057BT_nat ).

tff(sy_c_fAll_001t__VEBT____Definitions__OVEBT,type,
    fAll_VEBT_VEBT: fun_VEBT_VEBT_bool > bool ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_bool > real ).

tff(sy_c_fEx_001t__Nat__Onat,type,
    fEx_nat: fun_fu7168253649841883962l_bool ).

tff(sy_c_fEx_001t__Real__Oreal,type,
    fEx_real: fun_fu7511176317168184086l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: bool > fun_bool_bool ).

tff(sy_c_fdisj,type,
    fdisj: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Code____Numeral__Ointeger,type,
    fequal_Code_integer: fun_Co8310303586393106575r_bool ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1378794544173401837x_bool ).

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

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

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Rat__Orat,type,
    fequal_rat: fun_rat_fun_rat_bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fequal_001t__Set__Oset_It__Nat__Onat_J,type,
    fequal_set_nat: fun_se707272970763642173t_bool ).

tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
    fequal_VEBT_VEBT: fun_VE5877677516038742509T_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1841576699660343469l_bool ).

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

tff(sy_c_member_001t__HOL__Obool,type,
    member_bool: ( bool * set_bool ) > $o ).

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

tff(sy_c_member_001t__List__Olist_It__HOL__Obool_J,type,
    member_list_bool: ( list_bool * set_list_bool ) > $o ).

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

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

tff(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).

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

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

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: ( produc8923325533196201883nteger * set_Pr4811707699266497531nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    member7941544277139469958r_bool: ( produc7417142470700560111r_bool * set_Pr9123050470547338789r_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    member373505688050248522BT_nat: ( produc9072475918466114483BT_nat * set_Pr7556676689462069481BT_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_i____,type,
    i: 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 ).

% Relevant facts (10332)
tff(fact_0_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1)) ) ).

% not_min_Null_member
tff(fact_1_False,axiom,
    ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)))) ).

% False
tff(fact_2__092_060open_062_Iif_AminNull_A_ItreeList_A_B_Ahigh_Ami_An_J_Athen_Avebt__insert_Asummary_A_Ihigh_Ami_An_J_Aelse_Asummary_J_A_061_Asummary_092_060close_062,axiom,
    ( pp(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>(if minNull (treeList ! high mi n) then vebt_insert summary (high mi n) else summary) = summary\<close>
tff(fact_3__C8_C,axiom,
    na = m ).

% "8"
tff(fact_4__C12_C,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(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)),na)) ).

% "12"
tff(fact_5__C5_C,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_12)) ) ) ).

% "5"
tff(fact_6_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X2))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N))) ) ).

% in_children_def
tff(fact_7__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_8__C0_C,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(treeList)))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),na)) ) ).

% "0"
tff(fact_9_mimaxrel,axiom,
    ( ( xa != mi )
    & ( xa != ma ) ) ).

% mimaxrel
tff(fact_10__C11_C,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(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))))))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),na)) ) ).

% "11"
tff(fact_11_bit__split__inv,axiom,
    ! [X2: 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,X2),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),D)),D) = X2 ).

% bit_split_inv
tff(fact_12_abcdef,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)) ).

% abcdef
tff(fact_13__C3_C,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "3"
tff(fact_14_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,T),X2))
        | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,T),X2)) ) ) ).

% both_member_options_def
tff(fact_15__C1_C,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,summary),m)) ).

% "1"
tff(fact_16_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X2: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_17_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X2: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X2)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_18_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_19_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_20_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_21_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_22_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X2))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,Tree),X2))
          | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,Tree),X2)) ) ) ) ).

% member_valid_both_member_options
tff(fact_23_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_24_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_25_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_26_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_27_add__less__cancel__left,axiom,
    ! [C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_28_add__less__cancel__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_29_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_30_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_31_add__less__cancel__right,axiom,
    ! [A: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_32_add__less__cancel__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_33_add__le__cancel__left,axiom,
    ! [C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_34_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_35_add__le__cancel__left,axiom,
    ! [C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_36_add__le__cancel__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_37_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_38_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_39_add__le__cancel__right,axiom,
    ! [A: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_40_add__le__cancel__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_41_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2)) ) ) ).

% valid_member_both_member_options
tff(fact_42_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X2))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2)) ) ) ).

% both_member_options_equiv_member
tff(fact_43_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,vEBT_vebt_buildup(N)),X2)) ).

% buildup_nothing_in_leaf
tff(fact_44_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2)) ) ).

% min_Null_member
tff(fact_45_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,vEBT_vebt_buildup(N)),X2)) ).

% buildup_nothing_in_min_max
tff(fact_46_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_47_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_48_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_49_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_50_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_51_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_52_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_53_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_54_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),I,Y) = list_u1324408373059187874T_VEBT(Xs,I,Y) ).

% list_update_overwrite
tff(fact_55_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X2))
      <=> pp(aa_set_nat_bool(member_nat(X2),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_56_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B2))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
         => pp(aa_set_set_nat_bool(member_set_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_57_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B2))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
         => pp(aa_set_real_bool(member_real(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_58_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B2))
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),B2)) ) ) ).

% subset_code(1)
tff(fact_59_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),B2))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_60_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B2))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_61_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X2: set_nat,I: nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,X2))),A2)) ) ) ).

% set_update_subsetI
tff(fact_62_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X2: real,I: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A2))
     => ( pp(aa_set_real_bool(member_real(X2),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X2))),A2)) ) ) ).

% set_update_subsetI
tff(fact_63_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X2: nat,I: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(Xs)),A2))
     => ( pp(aa_set_nat_bool(member_nat(X2),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X2))),A2)) ) ) ).

% set_update_subsetI
tff(fact_64_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X2: vEBT_VEBT,I: nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X2))),A2)) ) ) ).

% set_update_subsetI
tff(fact_65_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X2: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A2))
     => ( pp(aa_set_int_bool(member_int(X2),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X2))),A2)) ) ) ).

% set_update_subsetI
tff(fact_66_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_67_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_68_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_69_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_70_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_71_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_72_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_73_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_74_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.left_commute
tff(fact_75_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_76_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_77_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_78_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_79_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_80_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_81_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_82_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_83_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_84_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_85_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_86_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_87_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_88_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_89_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_90_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_91_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_92_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),collect_VEBT_VEBT(P)))
    <=> pp(aa_VEBT_VEBT_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_93_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A),collect_real(P)))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_94_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> pp(aa_list_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_95_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),collect_set_nat(P)))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_96_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_97_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P)))
    <=> pp(aa_int_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_98_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_99_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_101_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_102_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_103_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_104_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(Q,X4)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_105_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
        <=> pp(aa_list_nat_bool(Q,X4)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_106_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
        <=> pp(aa_set_nat_bool(Q,X4)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_107_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
        <=> pp(aa_nat_bool(Q,X4)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_108_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(Q,X4)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_109_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_110_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_111_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_112_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_113_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_114_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_115_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_116_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_117_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_118_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_119_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_120_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_121_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_122_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_123_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_124_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_125_linorder__neqE__nat,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ) ).

% linorder_neqE_nat
tff(fact_126_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(P,N2))
         => ? [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
              & ~ pp(aa_nat_bool(P,M2)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_127_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_128_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_129_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_130_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_131_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_132_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_133_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => ? [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
            & ! [Y3: nat] :
                ( pp(aa_nat_bool(P,Y3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_134_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% nat_le_linear
tff(fact_135_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_136_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% eq_imp_le
tff(fact_137_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_138_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_139_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( I != I2 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),I2,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X5),I,X2) ) ) ).

% list_update_swap
tff(fact_140_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_141_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_142_add__le__imp__le__right,axiom,
    ! [A: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_143_add__le__imp__le__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_144_add__le__imp__le__left,axiom,
    ! [C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_145_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_146_add__le__imp__le__left,axiom,
    ! [C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_147_add__le__imp__le__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_148_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C2: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% le_iff_add
tff(fact_149_add__right__mono,axiom,
    ! [A: real,B: real,C: 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),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_right_mono
tff(fact_150_add__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C)),aa_rat_rat(plus_plus_rat(B),C))) ) ).

% add_right_mono
tff(fact_151_add__right__mono,axiom,
    ! [A: nat,B: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_right_mono
tff(fact_152_add__right__mono,axiom,
    ! [A: int,B: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_right_mono
tff(fact_153_less__eqE,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) ) ).

% less_eqE
tff(fact_154_add__left__mono,axiom,
    ! [A: real,B: real,C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_left_mono
tff(fact_155_add__left__mono,axiom,
    ! [A: rat,B: rat,C: 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(C),A)),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% add_left_mono
tff(fact_156_add__left__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_left_mono
tff(fact_157_add__left__mono,axiom,
    ! [A: int,B: int,C: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_left_mono
tff(fact_158_add__mono,axiom,
    ! [A: real,B: real,C: 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,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_mono
tff(fact_159_add__mono,axiom,
    ! [A: rat,B: rat,C: 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,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_mono
tff(fact_160_add__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_mono
tff(fact_161_add__mono,axiom,
    ! [A: int,B: int,C: 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,C),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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_mono
tff(fact_162_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_163_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_164_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_165_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),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,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_166_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_167_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_168_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),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,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_170_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_171_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_172_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K = 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,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_173_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = 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,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_174_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_175_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_176_add__less__imp__less__right,axiom,
    ! [A: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_177_add__less__imp__less__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_178_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_179_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_180_add__less__imp__less__left,axiom,
    ! [C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_181_add__less__imp__less__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_182_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: 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),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_strict_right_mono
tff(fact_183_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C)),aa_rat_rat(plus_plus_rat(B),C))) ) ).

% add_strict_right_mono
tff(fact_184_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_strict_right_mono
tff(fact_185_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_strict_right_mono
tff(fact_186_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_strict_left_mono
tff(fact_187_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: 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(C),A)),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% add_strict_left_mono
tff(fact_188_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_strict_left_mono
tff(fact_189_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_strict_left_mono
tff(fact_190_add__strict__mono,axiom,
    ! [A: real,B: real,C: 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,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_strict_mono
tff(fact_191_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: 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,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_strict_mono
tff(fact_192_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_strict_mono
tff(fact_193_add__strict__mono,axiom,
    ! [A: int,B: int,C: 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,C),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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_strict_mono
tff(fact_194_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_195_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_196_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_197_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_198_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_199_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_200_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_201_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_202_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_203_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_204_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_205_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_206_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: 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,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_207_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_208_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_209_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_210_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_211_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_212_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_213_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_less_add2
tff(fact_214_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_215_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_less_mono1
tff(fact_216_not__add__less2,axiom,
    ! [J: nat,I: 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),I)),I)) ).

% not_add_less2
tff(fact_217_not__add__less1,axiom,
    ! [I: 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,I),J)),I)) ).

% not_add_less1
tff(fact_218_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_219_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_220_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K2: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2) ) ).

% nat_le_iff_add
tff(fact_221_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_le_add2
tff(fact_222_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_le_add1
tff(fact_223_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),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,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_le_mono1
tff(fact_224_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_le_mono
tff(fact_225_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_226_add__leD2,axiom,
    ! [M: nat,K: 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,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% add_leD2
tff(fact_227_add__leD1,axiom,
    ! [M: nat,K: 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,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% add_leD1
tff(fact_228_le__add2,axiom,
    ! [N: nat,M: 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,M),N))) ).

% le_add2
tff(fact_229_le__add1,axiom,
    ! [N: nat,M: 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),M))) ).

% le_add1
tff(fact_230_add__leE,axiom,
    ! [M: nat,K: 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,M),K)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% add_leE
tff(fact_231_add__less__le__mono,axiom,
    ! [A: real,B: real,C: 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,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_less_le_mono
tff(fact_232_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: 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,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_less_le_mono
tff(fact_233_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_234_add__less__le__mono,axiom,
    ! [A: int,B: int,C: 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,C),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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_less_le_mono
tff(fact_235_add__le__less__mono,axiom,
    ! [A: real,B: real,C: 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,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_le_less_mono
tff(fact_236_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: 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,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_le_less_mono
tff(fact_237_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_238_add__le__less__mono,axiom,
    ! [A: int,B: int,C: 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,C),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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_le_less_mono
tff(fact_239_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_240_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_241_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_242_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_243_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_244_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_245_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_246_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_247_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N2))) )
     => 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,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_248_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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_249_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_250_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq2
tff(fact_251_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D)) ) ).

% valid_eq1
tff(fact_252_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq
tff(fact_253_dual__order_Orefl,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% dual_order.refl
tff(fact_254_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_255_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_256_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_257_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_258_order__refl,axiom,
    ! [X2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X2),X2)) ).

% order_refl
tff(fact_259_order__refl,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),X2)) ).

% order_refl
tff(fact_260_order__refl,axiom,
    ! [X2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),X2)) ).

% order_refl
tff(fact_261_order__refl,axiom,
    ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),X2)) ).

% order_refl
tff(fact_262_order__refl,axiom,
    ! [X2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),X2)) ).

% order_refl
tff(fact_263_inthall,axiom,
    ! [Xs: list_set_nat,P: 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(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,nth_set_nat(Xs,N))) ) ) ).

% inthall
tff(fact_264_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_265_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: 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(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_266_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_bool_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_267_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_268_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_int_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_269_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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_270_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
         => pp(aa_nat_bool(P,K3)) )
     => ( ! [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
           => ( ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),I4))
                 => pp(aa_nat_bool(P,I4)) )
             => pp(aa_nat_bool(P,K3)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_271_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_0_not
tff(fact_272_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_273_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_274_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_275_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_276_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_277_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y) = zero_zero_nat )
    <=> ( ( X2 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_304_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Y) )
    <=> ( ( X2 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_305_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_306_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_307_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_308_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_309_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_310_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_311_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_312_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_313_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_314_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_315_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_316_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_317_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_318_length__list__update,axiom,
    ! [Xs: list_bool,I: nat,X2: bool] : size_size_list_bool(list_update_bool(Xs,I,X2)) = size_size_list_bool(Xs) ).

% length_list_update
tff(fact_319_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] : size_size_list_nat(list_update_nat(Xs,I,X2)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_320_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X2: int] : size_size_list_int(list_update_int(Xs,I,X2)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_321_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_322_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_323_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_324_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_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_add__gr__0,axiom,
    ! [M: 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,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_366_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I))
     => ( list_u1324408373059187874T_VEBT(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_367_list__update__beyond,axiom,
    ! [Xs: list_bool,I: nat,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I))
     => ( list_update_bool(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_368_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I))
     => ( list_update_nat(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_369_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),I))
     => ( list_update_int(Xs,I,X2) = Xs ) ) ).

% list_update_beyond
tff(fact_370_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),I) = X2 ) ) ).

% nth_list_update_eq
tff(fact_371_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X2)),I))
      <=> pp(X2) ) ) ).

% nth_list_update_eq
tff(fact_372_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),I) = X2 ) ) ).

% nth_list_update_eq
tff(fact_373_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X2)),I) = X2 ) ) ).

% nth_list_update_eq
tff(fact_374_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),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,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_375_set__swap,axiom,
    ! [I: nat,Xs: list_bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),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,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_376_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),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,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_377_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),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,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_378_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_379_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_380_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_381_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_382_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = N ).

% Ex_list_of_length
tff(fact_383_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).

% Ex_list_of_length
tff(fact_384_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_385_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_386_zero__reorient,axiom,
    ! [X2: complex] :
      ( ( zero_zero_complex = X2 )
    <=> ( X2 = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_387_zero__reorient,axiom,
    ! [X2: real] :
      ( ( zero_zero_real = X2 )
    <=> ( X2 = zero_zero_real ) ) ).

% zero_reorient
tff(fact_388_zero__reorient,axiom,
    ! [X2: rat] :
      ( ( zero_zero_rat = X2 )
    <=> ( X2 = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_389_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
    <=> ( X2 = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_390_zero__reorient,axiom,
    ! [X2: int] :
      ( ( zero_zero_int = X2 )
    <=> ( X2 = zero_zero_int ) ) ).

% zero_reorient
tff(fact_391_size__neq__size__imp__neq,axiom,
    ! [X2: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X2) != size_s6755466524823107622T_VEBT(Y) )
     => ( X2 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_392_size__neq__size__imp__neq,axiom,
    ! [X2: list_bool,Y: list_bool] :
      ( ( size_size_list_bool(X2) != size_size_list_bool(Y) )
     => ( X2 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_393_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X2) != size_size_list_nat(Y) )
     => ( X2 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_394_size__neq__size__imp__neq,axiom,
    ! [X2: list_int,Y: list_int] :
      ( ( size_size_list_int(X2) != size_size_list_int(Y) )
     => ( X2 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_395_size__neq__size__imp__neq,axiom,
    ! [X2: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X2) != aa_num_nat(size_size_num,Y) )
     => ( X2 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_396_length__pos__if__in__set,axiom,
    ! [X2: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X2),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_397_length__pos__if__in__set,axiom,
    ! [X2: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X2),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_398_length__pos__if__in__set,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),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_399_length__pos__if__in__set,axiom,
    ! [X2: bool,Xs: list_bool] :
      ( member_bool(X2,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_400_length__pos__if__in__set,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),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_401_length__pos__if__in__set,axiom,
    ! [X2: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X2),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_402_length__induct,axiom,
    ! [P: 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(P,Ys2)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_403_length__induct,axiom,
    ! [P: 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(P,Ys2)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_404_length__induct,axiom,
    ! [P: 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(P,Ys2)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_405_length__induct,axiom,
    ! [P: 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(P,Ys2)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_406_zero__le,axiom,
    ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2)) ).

% zero_le
tff(fact_407_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_408_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_409_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_410_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_411_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_412_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_413_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_414_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_415_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_416_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_417_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_418_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_419_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_420_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_421_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_422_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_423_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_424_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_425_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_426_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_427_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_428_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_429_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_430_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_431_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( ~ pp(aa_nat_bool(P,N2))
             => ? [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
                  & ~ pp(aa_nat_bool(P,M2)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_432_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_433_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_434_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_435_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_436_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_437_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_438_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_439_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_440_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_441_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_442_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
         => ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I5),X_13)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I5),nth_VEBT_VEBT(Xs3,I5))) ) ) ) ).

% Skolem_list_nth
tff(fact_443_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
         => ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I5),X_13)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K )
          & ! [I5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I5),aa_nat_bool(nth_bool(Xs3),I5))) ) ) ) ).

% Skolem_list_nth
tff(fact_444_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I5),X_13)) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I5),aa_nat_nat(nth_nat(Xs3),I5))) ) ) ) ).

% Skolem_list_nth
tff(fact_445_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
         => ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I5),X_13)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I5),aa_nat_int(nth_int(Xs3),I5))) ) ) ) ).

% Skolem_list_nth
tff(fact_446_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I5) = nth_VEBT_VEBT(Ys,I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_447_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        & ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I5))
            <=> pp(aa_nat_bool(nth_bool(Ys),I5)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_448_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_449_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I5) = aa_nat_int(nth_int(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_450_add__decreasing,axiom,
    ! [A: real,C: 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,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_451_add__decreasing,axiom,
    ! [A: rat,C: 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,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B)) ) ) ).

% add_decreasing
tff(fact_452_add__decreasing,axiom,
    ! [A: nat,C: 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,C),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),C)),B)) ) ) ).

% add_decreasing
tff(fact_453_add__decreasing,axiom,
    ! [A: int,C: 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,C),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),C)),B)) ) ) ).

% add_decreasing
tff(fact_454_add__increasing,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing
tff(fact_455_add__increasing,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_increasing
tff(fact_456_add__increasing,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => 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),C))) ) ) ).

% add_increasing
tff(fact_457_add__increasing,axiom,
    ! [A: int,B: int,C: 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),C))
       => 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),C))) ) ) ).

% add_increasing
tff(fact_458_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),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),C)),B)) ) ) ).

% add_decreasing2
tff(fact_459_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),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),C)),B)) ) ) ).

% add_decreasing2
tff(fact_460_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),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),C)),B)) ) ) ).

% add_decreasing2
tff(fact_461_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),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),C)),B)) ) ) ).

% add_decreasing2
tff(fact_462_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( 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),C))) ) ) ).

% add_increasing2
tff(fact_463_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
     => ( 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),C))) ) ) ).

% add_increasing2
tff(fact_464_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( 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),C))) ) ) ).

% add_increasing2
tff(fact_465_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( 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),C))) ) ) ).

% add_increasing2
tff(fact_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_add__nonneg__eq__0__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y) = zero_zero_real )
        <=> ( ( X2 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_475_add__nonneg__eq__0__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_rat_rat(plus_plus_rat(X2),Y) = zero_zero_rat )
        <=> ( ( X2 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_476_add__nonneg__eq__0__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2))
     => ( 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,X2),Y) = zero_zero_nat )
        <=> ( ( X2 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_477_add__nonneg__eq__0__iff,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( 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,X2),Y) = zero_zero_int )
        <=> ( ( X2 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_478_add__nonpos__eq__0__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2),Y) = zero_zero_real )
        <=> ( ( X2 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_479_add__nonpos__eq__0__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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(X2),Y) = zero_zero_rat )
        <=> ( ( X2 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_480_add__nonpos__eq__0__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),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,X2),Y) = zero_zero_nat )
        <=> ( ( X2 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_481_add__nonpos__eq__0__iff,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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,X2),Y) = zero_zero_int )
        <=> ( ( X2 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_482_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_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_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))
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_491_pos__add__strict,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% pos_add_strict
tff(fact_492_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% pos_add_strict
tff(fact_493_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => 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),C))) ) ) ).

% pos_add_strict
tff(fact_494_pos__add__strict,axiom,
    ! [A: int,B: int,C: 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),C))
       => 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),C))) ) ) ).

% pos_add_strict
tff(fact_495_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,K3)) ) ) ) ).

% ex_least_nat_le
tff(fact_496_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_497_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_498_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_499_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_500_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_501_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_502_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_503_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: 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(P,X4)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_504_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P: 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(P,X4)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_505_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: 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(P,X4)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_506_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: 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(P,X4)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_507_in__set__conv__nth,axiom,
    ! [X2: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X2),set_set_nat2(Xs)))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s3254054031482475050et_nat(Xs)))
          & ( nth_set_nat(Xs,I5) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_508_in__set__conv__nth,axiom,
    ! [X2: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I5) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_509_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I5) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_510_in__set__conv__nth,axiom,
    ! [X2: bool,Xs: list_bool] :
      ( member_bool(X2,set_bool2(Xs))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I5))
          <=> pp(X2) ) ) ) ).

% in_set_conv_nth
tff(fact_511_in__set__conv__nth,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I5) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_512_in__set__conv__nth,axiom,
    ! [X2: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I5) = X2 ) ) ) ).

% in_set_conv_nth
tff(fact_513_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X2: 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(P,nth_set_nat(Xs,I3))) )
     => ( pp(aa_set_set_nat_bool(member_set_nat(X2),set_set_nat2(Xs)))
       => pp(aa_set_nat_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_514_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X2: 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(P,aa_nat_real(nth_real(Xs),I3))) )
     => ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
       => pp(aa_real_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_515_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X2: 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(P,nth_VEBT_VEBT(Xs,I3))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_516_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X2: 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(P,aa_nat_bool(nth_bool(Xs),I3))) )
     => ( member_bool(X2,set_bool2(Xs))
       => pp(aa_bool_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_517_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X2: 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(P,aa_nat_nat(nth_nat(Xs),I3))) )
     => ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_518_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X2: 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(P,aa_nat_int(nth_int(Xs),I3))) )
     => ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
       => pp(aa_int_bool(P,X2)) ) ) ).

% all_nth_imp_all_set
tff(fact_519_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I5))) ) ) ).

% all_set_conv_all_nth
tff(fact_520_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => pp(aa_bool_bool(P,X3)) )
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I5))) ) ) ).

% all_set_conv_all_nth
tff(fact_521_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I5))) ) ) ).

% all_set_conv_all_nth
tff(fact_522_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_int_bool(P,X3)) )
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I5))) ) ) ).

% all_set_conv_all_nth
tff(fact_523_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_524_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_525_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_526_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_527_le__cases3,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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),X2))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Z)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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),X2)) )
           => ( ( 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),X2)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X2))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_528_le__cases3,axiom,
    ! [X2: num,Y: num,Z: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),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),X2))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Z)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),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),X2)) )
           => ( ( 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),X2)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X2))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_529_le__cases3,axiom,
    ! [X2: nat,Y: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),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),X2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),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),X2)) )
           => ( ( 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),X2)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X2))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_530_le__cases3,axiom,
    ! [X2: int,Y: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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),X2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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),X2)) )
           => ( ( 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),X2)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X2))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_531_order__class_Oorder__eq__iff,axiom,
    ! [X2: set_int,Y: set_int] :
      ( ( X2 = Y )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_532_order__class_Oorder__eq__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( ( X2 = Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_533_order__class_Oorder__eq__iff,axiom,
    ! [X2: num,Y: num] :
      ( ( X2 = Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_534_order__class_Oorder__eq__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_535_order__class_Oorder__eq__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2)) ) ) ).

% order_class.order_eq_iff
tff(fact_536_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_537_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_538_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_539_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_540_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_541_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( ( B = C )
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_542_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_543_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_544_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_545_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_546_order__antisym,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X2))
       => ( X2 = Y ) ) ) ).

% order_antisym
tff(fact_547_order__antisym,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2))
       => ( X2 = Y ) ) ) ).

% order_antisym
tff(fact_548_order__antisym,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2))
       => ( X2 = Y ) ) ) ).

% order_antisym
tff(fact_549_order__antisym,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2))
       => ( X2 = Y ) ) ) ).

% order_antisym
tff(fact_550_order__antisym,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2))
       => ( X2 = Y ) ) ) ).

% order_antisym
tff(fact_551_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: 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),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% order.trans
tff(fact_552_order_Otrans,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% order.trans
tff(fact_553_order_Otrans,axiom,
    ! [A: num,B: num,C: 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),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% order.trans
tff(fact_554_order_Otrans,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% order.trans
tff(fact_555_order_Otrans,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% order.trans
tff(fact_556_order__trans,axiom,
    ! [X2: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X2),Z)) ) ) ).

% order_trans
tff(fact_557_order__trans,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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,X2),Z)) ) ) ).

% order_trans
tff(fact_558_order__trans,axiom,
    ! [X2: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),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,X2),Z)) ) ) ).

% order_trans
tff(fact_559_order__trans,axiom,
    ! [X2: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),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,X2),Z)) ) ) ).

% order_trans
tff(fact_560_order__trans,axiom,
    ! [X2: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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,X2),Z)) ) ) ).

% order_trans
tff(fact_561_linorder__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: rat,B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_562_linorder__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: num,B3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
           => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_563_linorder__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_564_linorder__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_565_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_566_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_567_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_568_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_569_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_570_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_571_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_572_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_573_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_574_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_575_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C),A)) ) ) ).

% dual_order.trans
tff(fact_576_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: 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,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% dual_order.trans
tff(fact_577_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: 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,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% dual_order.trans
tff(fact_578_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: 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,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% dual_order.trans
tff(fact_579_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: 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,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% dual_order.trans
tff(fact_580_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_581_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_582_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_583_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_584_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_585_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_586_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_587_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_588_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_589_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_590_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_591_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_592_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: 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),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_593_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: 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),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_594_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_595_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_596_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: 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),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_597_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: 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),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_598_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_599_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_600_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_601_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_602_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_603_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_604_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_605_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_606_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_607_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_608_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: 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)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_609_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: 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)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_610_order__eq__refl,axiom,
    ! [X2: set_int,Y: set_int] :
      ( ( X2 = Y )
     => pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y)) ) ).

% order_eq_refl
tff(fact_611_order__eq__refl,axiom,
    ! [X2: rat,Y: rat] :
      ( ( X2 = Y )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ).

% order_eq_refl
tff(fact_612_order__eq__refl,axiom,
    ! [X2: num,Y: num] :
      ( ( X2 = Y )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y)) ) ).

% order_eq_refl
tff(fact_613_order__eq__refl,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ).

% order_eq_refl
tff(fact_614_order__eq__refl,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y)) ) ).

% order_eq_refl
tff(fact_615_linorder__linear,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2)) ) ).

% linorder_linear
tff(fact_616_linorder__linear,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2)) ) ).

% linorder_linear
tff(fact_617_linorder__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2)) ) ).

% linorder_linear
tff(fact_618_linorder__linear,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2)) ) ).

% linorder_linear
tff(fact_619_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_620_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_621_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_622_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_623_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_624_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_625_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_626_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_627_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_628_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_629_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_630_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_631_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_632_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_633_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_634_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_635_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_636_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_637_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_638_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_639_linorder__le__cases,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2)) ) ).

% linorder_le_cases
tff(fact_640_linorder__le__cases,axiom,
    ! [X2: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2)) ) ).

% linorder_le_cases
tff(fact_641_linorder__le__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2)) ) ).

% linorder_le_cases
tff(fact_642_linorder__le__cases,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2)) ) ).

% linorder_le_cases
tff(fact_643_order__antisym__conv,axiom,
    ! [Y: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
      <=> ( X2 = Y ) ) ) ).

% order_antisym_conv
tff(fact_644_order__antisym__conv,axiom,
    ! [Y: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% order_antisym_conv
tff(fact_645_order__antisym__conv,axiom,
    ! [Y: num,X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% order_antisym_conv
tff(fact_646_order__antisym__conv,axiom,
    ! [Y: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% order_antisym_conv
tff(fact_647_order__antisym__conv,axiom,
    ! [Y: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% order_antisym_conv
tff(fact_648_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X2: 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(X2),set_set_nat2(list_update_set_nat(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_649_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X2: 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(X2),set_real2(list_update_real(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_650_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X2: 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,X2),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_651_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(X2,set_bool2(list_update_bool(Xs,N,X2))) ) ).

% set_update_memI
tff(fact_652_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X2: 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(X2),set_nat2(list_update_nat(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_653_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X2: 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(X2),set_int2(list_update_int(Xs,N,X2)))) ) ).

% set_update_memI
tff(fact_654_lt__ex,axiom,
    ! [X2: real] :
    ? [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X2)) ).

% lt_ex
tff(fact_655_lt__ex,axiom,
    ! [X2: rat] :
    ? [Y2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y2),X2)) ).

% lt_ex
tff(fact_656_lt__ex,axiom,
    ! [X2: int] :
    ? [Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y2),X2)) ).

% lt_ex
tff(fact_657_gt__ex,axiom,
    ! [X2: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X_1)) ).

% gt_ex
tff(fact_658_gt__ex,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),X_1)) ).

% gt_ex
tff(fact_659_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),X_1)) ).

% gt_ex
tff(fact_660_gt__ex,axiom,
    ! [X2: int] :
    ? [X_1: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),X_1)) ).

% gt_ex
tff(fact_661_dense,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),Y)) ) ) ).

% dense
tff(fact_662_dense,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ? [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),Y)) ) ) ).

% dense
tff(fact_663_less__imp__neq,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( X2 != Y ) ) ).

% less_imp_neq
tff(fact_664_less__imp__neq,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ( X2 != Y ) ) ).

% less_imp_neq
tff(fact_665_less__imp__neq,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ( X2 != Y ) ) ).

% less_imp_neq
tff(fact_666_less__imp__neq,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ( X2 != Y ) ) ).

% less_imp_neq
tff(fact_667_less__imp__neq,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ( X2 != Y ) ) ).

% less_imp_neq
tff(fact_668_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_669_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_670_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_671_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_672_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_673_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_674_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_675_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_676_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_677_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_678_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_679_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_680_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_681_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_682_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_683_less__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X4))
             => pp(aa_nat_bool(P,Y3)) )
         => pp(aa_nat_bool(P,X4)) )
     => pp(aa_nat_bool(P,A)) ) ).

% less_induct
tff(fact_684_antisym__conv3,axiom,
    ! [Y: real,X2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv3
tff(fact_685_antisym__conv3,axiom,
    ! [Y: rat,X2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv3
tff(fact_686_antisym__conv3,axiom,
    ! [Y: num,X2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv3
tff(fact_687_antisym__conv3,axiom,
    ! [Y: nat,X2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv3
tff(fact_688_antisym__conv3,axiom,
    ! [Y: int,X2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv3
tff(fact_689_linorder__cases,axiom,
    ! [X2: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( ( X2 != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ) ).

% linorder_cases
tff(fact_690_linorder__cases,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ( ( X2 != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ) ).

% linorder_cases
tff(fact_691_linorder__cases,axiom,
    ! [X2: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ( ( X2 != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ) ).

% linorder_cases
tff(fact_692_linorder__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ( ( X2 != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ) ).

% linorder_cases
tff(fact_693_linorder__cases,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ( ( X2 != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ) ).

% linorder_cases
tff(fact_694_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_695_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_696_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_697_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_698_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_699_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_700_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_701_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_702_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_703_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_704_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [N3: nat] :
          ( pp(aa_nat_bool(P,N3))
          & ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N3))
             => ~ pp(aa_nat_bool(P,M4)) ) ) ) ).

% exists_least_iff
tff(fact_705_linorder__less__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),A3))
       => ( ! [A3: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_706_linorder__less__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),A3))
       => ( ! [A3: rat,B3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_707_linorder__less__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A3),A3))
       => ( ! [A3: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_708_linorder__less__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_709_linorder__less__wlog,axiom,
    ! [P: 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(P,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_710_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans
tff(fact_711_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans
tff(fact_712_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: 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),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans
tff(fact_713_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans
tff(fact_714_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans
tff(fact_715_not__less__iff__gr__or__eq,axiom,
    ! [X2: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2))
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_716_not__less__iff__gr__or__eq,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2))
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_717_not__less__iff__gr__or__eq,axiom,
    ! [X2: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2))
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_718_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2))
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_719_not__less__iff__gr__or__eq,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2))
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_720_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_721_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: 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,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_722_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: 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,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_723_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: 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,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_724_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: 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,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_725_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_726_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_727_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_728_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_729_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_730_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_731_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_732_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_733_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_734_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_735_linorder__neqE,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ) ).

% linorder_neqE
tff(fact_736_linorder__neqE,axiom,
    ! [X2: rat,Y: rat] :
      ( ( X2 != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ) ).

% linorder_neqE
tff(fact_737_linorder__neqE,axiom,
    ! [X2: num,Y: num] :
      ( ( X2 != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ) ).

% linorder_neqE
tff(fact_738_linorder__neqE,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ) ).

% linorder_neqE
tff(fact_739_linorder__neqE,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ) ).

% linorder_neqE
tff(fact_740_order__less__asym,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ).

% order_less_asym
tff(fact_741_order__less__asym,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ).

% order_less_asym
tff(fact_742_order__less__asym,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ).

% order_less_asym
tff(fact_743_order__less__asym,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ).

% order_less_asym
tff(fact_744_order__less__asym,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ).

% order_less_asym
tff(fact_745_linorder__neq__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ) ).

% linorder_neq_iff
tff(fact_746_linorder__neq__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( ( X2 != Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ) ).

% linorder_neq_iff
tff(fact_747_linorder__neq__iff,axiom,
    ! [X2: num,Y: num] :
      ( ( X2 != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ) ).

% linorder_neq_iff
tff(fact_748_linorder__neq__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ) ).

% linorder_neq_iff
tff(fact_749_linorder__neq__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ) ).

% linorder_neq_iff
tff(fact_750_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_751_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_752_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_753_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_754_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_755_order__less__trans,axiom,
    ! [X2: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z)) ) ) ).

% order_less_trans
tff(fact_756_order__less__trans,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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,X2),Z)) ) ) ).

% order_less_trans
tff(fact_757_order__less__trans,axiom,
    ! [X2: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),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,X2),Z)) ) ) ).

% order_less_trans
tff(fact_758_order__less__trans,axiom,
    ! [X2: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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,X2),Z)) ) ) ).

% order_less_trans
tff(fact_759_order__less__trans,axiom,
    ! [X2: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),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,X2),Z)) ) ) ).

% order_less_trans
tff(fact_760_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_761_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_762_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_763_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_764_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_765_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_766_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_767_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_768_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_769_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_770_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_771_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_772_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_773_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_774_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_775_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_776_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_777_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_778_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_779_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_780_order__less__irrefl,axiom,
    ! [X2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X2)) ).

% order_less_irrefl
tff(fact_781_order__less__irrefl,axiom,
    ! [X2: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),X2)) ).

% order_less_irrefl
tff(fact_782_order__less__irrefl,axiom,
    ! [X2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),X2)) ).

% order_less_irrefl
tff(fact_783_order__less__irrefl,axiom,
    ! [X2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),X2)) ).

% order_less_irrefl
tff(fact_784_order__less__irrefl,axiom,
    ! [X2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),X2)) ).

% order_less_irrefl
tff(fact_785_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: 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),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_786_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_787_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_788_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: 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),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_789_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: 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),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_790_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: 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),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_791_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_792_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_793_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: 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),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_794_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: 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),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_795_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: 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)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_796_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: 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)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_797_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: 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)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_798_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: 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)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_799_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: 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)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_800_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_801_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_802_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_803_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_804_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_805_order__less__not__sym,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ).

% order_less_not_sym
tff(fact_806_order__less__not__sym,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ).

% order_less_not_sym
tff(fact_807_order__less__not__sym,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ).

% order_less_not_sym
tff(fact_808_order__less__not__sym,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ).

% order_less_not_sym
tff(fact_809_order__less__not__sym,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ).

% order_less_not_sym
tff(fact_810_order__less__imp__triv,axiom,
    ! [X2: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_811_order__less__imp__triv,axiom,
    ! [X2: rat,Y: rat,P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_812_order__less__imp__triv,axiom,
    ! [X2: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_813_order__less__imp__triv,axiom,
    ! [X2: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_814_order__less__imp__triv,axiom,
    ! [X2: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_815_linorder__less__linear,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
      | ( X2 = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ).

% linorder_less_linear
tff(fact_816_linorder__less__linear,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
      | ( X2 = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ).

% linorder_less_linear
tff(fact_817_linorder__less__linear,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
      | ( X2 = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ).

% linorder_less_linear
tff(fact_818_linorder__less__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
      | ( X2 = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ).

% linorder_less_linear
tff(fact_819_linorder__less__linear,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
      | ( X2 = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ).

% linorder_less_linear
tff(fact_820_order__less__imp__not__eq,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_821_order__less__imp__not__eq,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_822_order__less__imp__not__eq,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_823_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_824_order__less__imp__not__eq,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_825_order__less__imp__not__eq2,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_826_order__less__imp__not__eq2,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_827_order__less__imp__not__eq2,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_828_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_829_order__less__imp__not__eq2,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
tff(fact_830_order__less__imp__not__less,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ).

% order_less_imp_not_less
tff(fact_831_order__less__imp__not__less,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ).

% order_less_imp_not_less
tff(fact_832_order__less__imp__not__less,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ).

% order_less_imp_not_less
tff(fact_833_order__less__imp__not__less,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ).

% order_less_imp_not_less
tff(fact_834_order__less__imp__not__less,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ).

% order_less_imp_not_less
tff(fact_835_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),J) = X2 ) )
        & ( ( I != J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X2),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_836_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X2: bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X2)),J))
      <=> ( ( ( I = J )
           => pp(X2) )
          & ( ( I != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_837_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),J) = X2 ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X2)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_838_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X2)),J) = X2 ) )
        & ( ( I != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X2)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_839_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X2) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X2 ) ) ) ).

% list_update_same_conv
tff(fact_840_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_bool,X2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I,X2) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I))
        <=> pp(X2) ) ) ) ).

% list_update_same_conv
tff(fact_841_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I,X2) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X2 ) ) ) ).

% list_update_same_conv
tff(fact_842_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( list_update_int(Xs,I,X2) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X2 ) ) ) ).

% list_update_same_conv
tff(fact_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_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_859_add__strict__increasing,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing
tff(fact_860_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_strict_increasing
tff(fact_861_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => 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),C))) ) ) ).

% add_strict_increasing
tff(fact_862_add__strict__increasing,axiom,
    ! [A: int,B: int,C: 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),C))
       => 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),C))) ) ) ).

% add_strict_increasing
tff(fact_863_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_864_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_strict_increasing2
tff(fact_865_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => 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),C))) ) ) ).

% add_strict_increasing2
tff(fact_866_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: 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),C))
       => 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),C))) ) ) ).

% add_strict_increasing2
tff(fact_867_leD,axiom,
    ! [Y: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% leD
tff(fact_868_leD,axiom,
    ! [Y: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X2))
     => ~ pp(aa_set_int_bool(ord_less_set_int(X2),Y)) ) ).

% leD
tff(fact_869_leD,axiom,
    ! [Y: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y)) ) ).

% leD
tff(fact_870_leD,axiom,
    ! [Y: num,X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y)) ) ).

% leD
tff(fact_871_leD,axiom,
    ! [Y: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y)) ) ).

% leD
tff(fact_872_leD,axiom,
    ! [Y: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y)) ) ).

% leD
tff(fact_873_leI,axiom,
    ! [X2: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X2)) ) ).

% leI
tff(fact_874_leI,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2)) ) ).

% leI
tff(fact_875_leI,axiom,
    ! [X2: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2)) ) ).

% leI
tff(fact_876_leI,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2)) ) ).

% leI
tff(fact_877_leI,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2)) ) ).

% leI
tff(fact_878_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_879_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_880_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_881_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_882_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_883_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_884_antisym__conv1,axiom,
    ! [X2: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv1
tff(fact_885_antisym__conv1,axiom,
    ! [X2: set_int,Y: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(X2),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv1
tff(fact_886_antisym__conv1,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv1
tff(fact_887_antisym__conv1,axiom,
    ! [X2: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv1
tff(fact_888_antisym__conv1,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv1
tff(fact_889_antisym__conv1,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv1
tff(fact_890_antisym__conv2,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv2
tff(fact_891_antisym__conv2,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
     => ( ~ pp(aa_set_int_bool(ord_less_set_int(X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv2
tff(fact_892_antisym__conv2,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv2
tff(fact_893_antisym__conv2,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv2
tff(fact_894_antisym__conv2,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv2
tff(fact_895_antisym__conv2,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
      <=> ( X2 = Y ) ) ) ).

% antisym_conv2
tff(fact_896_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_897_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_898_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_899_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_900_less__le__not__le,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X2)) ) ) ).

% less_le_not_le
tff(fact_901_less__le__not__le,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(Y),X2)) ) ) ).

% less_le_not_le
tff(fact_902_less__le__not__le,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2)) ) ) ).

% less_le_not_le
tff(fact_903_less__le__not__le,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2)) ) ) ).

% less_le_not_le
tff(fact_904_less__le__not__le,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2)) ) ) ).

% less_le_not_le
tff(fact_905_less__le__not__le,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2)) ) ) ).

% less_le_not_le
tff(fact_906_not__le__imp__less,axiom,
    ! [Y: real,X2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% not_le_imp_less
tff(fact_907_not__le__imp__less,axiom,
    ! [Y: rat,X2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y)) ) ).

% not_le_imp_less
tff(fact_908_not__le__imp__less,axiom,
    ! [Y: num,X2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y)) ) ).

% not_le_imp_less
tff(fact_909_not__le__imp__less,axiom,
    ! [Y: nat,X2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y)) ) ).

% not_le_imp_less
tff(fact_910_not__le__imp__less,axiom,
    ! [Y: int,X2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y)) ) ).

% not_le_imp_less
tff(fact_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans1
tff(fact_924_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% order.strict_trans1
tff(fact_925_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_926_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: 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),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans1
tff(fact_927_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_928_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans1
tff(fact_929_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans2
tff(fact_930_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% order.strict_trans2
tff(fact_931_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_932_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: 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),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans2
tff(fact_933_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_934_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans2
tff(fact_935_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_936_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_937_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_938_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_939_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_940_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_941_dense__ge__bounded,axiom,
    ! [Z: real,X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( ! [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),X2))
             => 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_942_dense__ge__bounded,axiom,
    ! [Z: rat,X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ( ! [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),X2))
             => 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_943_dense__le__bounded,axiom,
    ! [X2: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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_944_dense__le__bounded,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_958_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_959_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: 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,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_960_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: 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,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_961_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: 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,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_962_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: 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,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_963_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_964_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_965_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: 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,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_966_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: 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,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_967_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: 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,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_968_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: 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,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_order__le__less,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_less
tff(fact_988_order__le__less,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_less
tff(fact_989_order__le__less,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_less
tff(fact_990_order__le__less,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_less
tff(fact_991_order__le__less,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_less
tff(fact_992_order__le__less,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_less
tff(fact_993_order__less__le,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
        & ( X2 != Y ) ) ) ).

% order_less_le
tff(fact_994_order__less__le,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
        & ( X2 != Y ) ) ) ).

% order_less_le
tff(fact_995_order__less__le,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
        & ( X2 != Y ) ) ) ).

% order_less_le
tff(fact_996_order__less__le,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
        & ( X2 != Y ) ) ) ).

% order_less_le
tff(fact_997_order__less__le,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
        & ( X2 != Y ) ) ) ).

% order_less_le
tff(fact_998_order__less__le,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
        & ( X2 != Y ) ) ) ).

% order_less_le
tff(fact_999_linorder__not__le,axiom,
    ! [X2: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ).

% linorder_not_le
tff(fact_1000_linorder__not__le,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ).

% linorder_not_le
tff(fact_1001_linorder__not__le,axiom,
    ! [X2: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ).

% linorder_not_le
tff(fact_1002_linorder__not__le,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ).

% linorder_not_le
tff(fact_1003_linorder__not__le,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ).

% linorder_not_le
tff(fact_1004_linorder__not__less,axiom,
    ! [X2: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X2)) ) ).

% linorder_not_less
tff(fact_1005_linorder__not__less,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2)) ) ).

% linorder_not_less
tff(fact_1006_linorder__not__less,axiom,
    ! [X2: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2)) ) ).

% linorder_not_less
tff(fact_1007_linorder__not__less,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2)) ) ).

% linorder_not_less
tff(fact_1008_linorder__not__less,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2)) ) ).

% linorder_not_less
tff(fact_1009_order__less__imp__le,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ).

% order_less_imp_le
tff(fact_1010_order__less__imp__le,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y)) ) ).

% order_less_imp_le
tff(fact_1011_order__less__imp__le,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ).

% order_less_imp_le
tff(fact_1012_order__less__imp__le,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y)) ) ).

% order_less_imp_le
tff(fact_1013_order__less__imp__le,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ).

% order_less_imp_le
tff(fact_1014_order__less__imp__le,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y)) ) ).

% order_less_imp_le
tff(fact_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_order__le__less__trans,axiom,
    ! [X2: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_1028_order__le__less__trans,axiom,
    ! [X2: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_1029_order__le__less__trans,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_1030_order__le__less__trans,axiom,
    ! [X2: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),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,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_1031_order__le__less__trans,axiom,
    ! [X2: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),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,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_1032_order__le__less__trans,axiom,
    ! [X2: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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,X2),Z)) ) ) ).

% order_le_less_trans
tff(fact_1033_order__less__le__trans,axiom,
    ! [X2: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_1034_order__less__le__trans,axiom,
    ! [X2: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X2),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_1035_order__less__le__trans,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_1036_order__less__le__trans,axiom,
    ! [X2: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),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,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_1037_order__less__le__trans,axiom,
    ! [X2: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_1038_order__less__le__trans,axiom,
    ! [X2: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),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,X2),Z)) ) ) ).

% order_less_le_trans
tff(fact_1039_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: 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),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1040_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1041_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1042_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: 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),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1043_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: 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),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1044_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: 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),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1045_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1046_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1047_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: 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),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1048_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: 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),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1049_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1050_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1051_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1052_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1053_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1054_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1055_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1056_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1057_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1058_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1059_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1060_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1061_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1062_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1063_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: 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),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1064_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1065_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1066_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1067_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1068_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: 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),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1069_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: 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)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1070_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1071_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1072_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: 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)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1073_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: 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)),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1074_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: 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)),C))
       => ( ! [X4: real,Y2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1075_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: 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)),C))
       => ( ! [X4: rat,Y2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1076_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: 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)),C))
       => ( ! [X4: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1077_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: 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)),C))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1078_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: 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)),C))
       => ( ! [X4: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1079_linorder__le__less__linear,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ).

% linorder_le_less_linear
tff(fact_1080_linorder__le__less__linear,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ).

% linorder_le_less_linear
tff(fact_1081_linorder__le__less__linear,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X2)) ) ).

% linorder_le_less_linear
tff(fact_1082_linorder__le__less__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X2)) ) ).

% linorder_le_less_linear
tff(fact_1083_linorder__le__less__linear,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ).

% linorder_le_less_linear
tff(fact_1084_order__le__imp__less__or__eq,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1085_order__le__imp__less__or__eq,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1086_order__le__imp__less__or__eq,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1087_order__le__imp__less__or__eq,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1088_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1089_order__le__imp__less__or__eq,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1090_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_1091_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_1092_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_1093_field__le__epsilon,axiom,
    ! [X2: 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,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ).

% field_le_epsilon
tff(fact_1094_field__le__epsilon,axiom,
    ! [X2: 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,X2),aa_rat_rat(plus_plus_rat(Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ).

% field_le_epsilon
tff(fact_1095_add__less__zeroD,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_1096_add__less__zeroD,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(X2),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_1097_add__less__zeroD,axiom,
    ! [X2: 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,X2),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_1098_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1099_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_nat_bool(member_nat(X4),B2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ).

% subsetI
tff(fact_1100_subsetI,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),B2)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2)) ) ).

% subsetI
tff(fact_1101_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X4),B2)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ).

% subsetI
tff(fact_1102_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_set_real_bool(member_real(X4),B2)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ).

% subsetI
tff(fact_1103_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_set_int_bool(member_int(X4),B2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% subsetI
tff(fact_1104_psubsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_set_int_bool(ord_less_set_int(A2),B2)) ) ) ).

% psubsetI
tff(fact_1105_subset__antisym,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( A2 = B2 ) ) ) ).

% subset_antisym
tff(fact_1106_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_1107_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),zero_zero_nat))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_1108_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_1109_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_1110_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_1111_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_1112_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_1113_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1114_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_1115_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X3: list_nat] : ~ pp(aa_list_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_1123_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X3: set_nat] : ~ pp(aa_set_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_1124_empty__Collect__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X3: real] : ~ pp(aa_real_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_1125_empty__Collect__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X3: nat] : ~ pp(aa_nat_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_1126_empty__Collect__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X3: int] : ~ pp(aa_int_bool(P,X3)) ) ).

% empty_Collect_eq
tff(fact_1127_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X3: list_nat] : ~ pp(aa_list_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_1128_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X3: set_nat] : ~ pp(aa_set_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_1129_Collect__empty__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X3: real] : ~ pp(aa_real_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_1130_Collect__empty__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X3: nat] : ~ pp(aa_nat_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_1131_Collect__empty__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X3: int] : ~ pp(aa_int_bool(P,X3)) ) ).

% Collect_empty_eq
tff(fact_1132_all__not__in__conv,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
    <=> ( A2 = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
tff(fact_1133_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X3: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_1134_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X3: real] : ~ pp(aa_set_real_bool(member_real(X3),A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_1135_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X3: nat] : ~ pp(aa_set_nat_bool(member_nat(X3),A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_1136_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X3: int] : ~ pp(aa_set_int_bool(member_int(X3),A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_1137_empty__iff,axiom,
    ! [C: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),bot_bo8194388402131092736T_VEBT)) ).

% empty_iff
tff(fact_1138_empty__iff,axiom,
    ! [C: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(C),bot_bot_set_set_nat)) ).

% empty_iff
tff(fact_1139_empty__iff,axiom,
    ! [C: real] : ~ pp(aa_set_real_bool(member_real(C),bot_bot_set_real)) ).

% empty_iff
tff(fact_1140_empty__iff,axiom,
    ! [C: nat] : ~ pp(aa_set_nat_bool(member_nat(C),bot_bot_set_nat)) ).

% empty_iff
tff(fact_1141_empty__iff,axiom,
    ! [C: int] : ~ pp(aa_set_int_bool(member_int(C),bot_bot_set_int)) ).

% empty_iff
tff(fact_1142_empty__subsetI,axiom,
    ! [A2: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A2)) ).

% empty_subsetI
tff(fact_1143_empty__subsetI,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A2)) ).

% empty_subsetI
tff(fact_1144_empty__subsetI,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A2)) ).

% empty_subsetI
tff(fact_1145_subset__empty,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1146_subset__empty,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1147_subset__empty,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1148_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs))) ).

% List.finite_set
tff(fact_1149_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_1150_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_1151_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_1152_ex__in__conv,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ? [X3: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
    <=> ( A2 != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
tff(fact_1153_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X3: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1154_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X3: real] : pp(aa_set_real_bool(member_real(X3),A2))
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1155_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X3: nat] : pp(aa_set_nat_bool(member_nat(X3),A2))
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1156_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X3: int] : pp(aa_set_int_bool(member_int(X3),A2))
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1157_equals0I,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [Y2: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),A2))
     => ( A2 = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
tff(fact_1158_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y2: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(Y2),A2))
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1159_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y2: real] : ~ pp(aa_set_real_bool(member_real(Y2),A2))
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1160_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y2: nat] : ~ pp(aa_set_nat_bool(member_nat(Y2),A2))
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1161_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y2: int] : ~ pp(aa_set_int_bool(member_int(Y2),A2))
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1162_equals0D,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( A2 = bot_bo8194388402131092736T_VEBT )
     => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2)) ) ).

% equals0D
tff(fact_1163_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ).

% equals0D
tff(fact_1164_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ pp(aa_set_real_bool(member_real(A),A2)) ) ).

% equals0D
tff(fact_1165_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ pp(aa_set_nat_bool(member_nat(A),A2)) ) ).

% equals0D
tff(fact_1166_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ pp(aa_set_int_bool(member_int(A),A2)) ) ).

% equals0D
tff(fact_1167_emptyE,axiom,
    ! [A: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),bot_bo8194388402131092736T_VEBT)) ).

% emptyE
tff(fact_1168_emptyE,axiom,
    ! [A: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(A),bot_bot_set_set_nat)) ).

% emptyE
tff(fact_1169_emptyE,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(member_real(A),bot_bot_set_real)) ).

% emptyE
tff(fact_1170_emptyE,axiom,
    ! [A: nat] : ~ pp(aa_set_nat_bool(member_nat(A),bot_bot_set_nat)) ).

% emptyE
tff(fact_1171_emptyE,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(member_int(A),bot_bot_set_int)) ).

% emptyE
tff(fact_1172_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A2),bot_bot_set_real)) ).

% not_psubset_empty
tff(fact_1173_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_1174_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).

% not_psubset_empty
tff(fact_1175_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_1176_psubsetD,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),B2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
       => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ) ).

% psubsetD
tff(fact_1177_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% psubsetD
tff(fact_1178_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_1179_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
     => ( pp(aa_set_real_bool(member_real(C),A2))
       => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% psubsetD
tff(fact_1180_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_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_bot_Oextremum__uniqueI,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
     => ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1191_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_1192_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_1193_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_1194_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_1195_bot_Oextremum__unique,axiom,
    ! [A: filter_nat] :
      ( ord_le2510731241096832064er_nat(A,bot_bot_filter_nat)
    <=> ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_1196_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_1197_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_1198_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_1199_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_1200_bot_Oextremum,axiom,
    ! [A: filter_nat] : ord_le2510731241096832064er_nat(bot_bot_filter_nat,A) ).

% bot.extremum
tff(fact_1201_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_1202_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_1203_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_1204_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_1205_bot_Onot__eq__extremum,axiom,
    ! [A: filter_nat] :
      ( ( A != bot_bot_filter_nat )
    <=> ord_less_filter_nat(bot_bot_filter_nat,A) ) ).

% bot.not_eq_extremum
tff(fact_1206_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_1207_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_1208_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_1209_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_1210_bot_Oextremum__strict,axiom,
    ! [A: filter_nat] : ~ ord_less_filter_nat(A,bot_bot_filter_nat) ).

% bot.extremum_strict
tff(fact_1211_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_1212_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_1213_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_1214_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_1215_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_1216_finite__list,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_1217_finite__list,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_1218_finite__list,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_1219_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_1220_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_1221_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_1222_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_1223_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_1224_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_1225_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_1226_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_1227_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_1228_linorder__neqE__linordered__idom,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1229_linorder__neqE__linordered__idom,axiom,
    ! [X2: rat,Y: rat] :
      ( ( X2 != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X2)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1230_linorder__neqE__linordered__idom,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X2)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1231_linordered__field__no__ub,axiom,
    ! [X: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_1)) ).

% linordered_field_no_ub
tff(fact_1232_linordered__field__no__ub,axiom,
    ! [X: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_1)) ).

% linordered_field_no_ub
tff(fact_1233_linordered__field__no__lb,axiom,
    ! [X: real] :
    ? [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X)) ).

% linordered_field_no_lb
tff(fact_1234_linordered__field__no__lb,axiom,
    ! [X: rat] :
    ? [Y2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y2),X)) ).

% linordered_field_no_lb
tff(fact_1235_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_1236_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_1237_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_1238_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_1239_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_1240_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_1241_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_1242_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_1243_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1244_subset__psubset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_set_int(B2),C4))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C4)) ) ) ).

% subset_psubset_trans
tff(fact_1245_subset__not__subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ) ).

% subset_not_subset_eq
tff(fact_1246_psubset__subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C4)) ) ) ).

% psubset_subset_trans
tff(fact_1247_psubset__imp__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% psubset_imp_subset
tff(fact_1248_Collect__mono__iff,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q)))
    <=> ! [X3: real] :
          ( pp(aa_real_bool(P,X3))
         => pp(aa_real_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1249_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X3: list_nat] :
          ( pp(aa_list_nat_bool(P,X3))
         => pp(aa_list_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1250_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_nat_bool(P,X3))
         => pp(aa_set_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1251_Collect__mono__iff,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)))
    <=> ! [X3: nat] :
          ( pp(aa_nat_bool(P,X3))
         => pp(aa_nat_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1252_Collect__mono__iff,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q)))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(P,X3))
         => pp(aa_int_bool(Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_1253_set__eq__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ) ).

% set_eq_subset
tff(fact_1254_subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4)) ) ) ).

% subset_trans
tff(fact_1255_Collect__mono,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
         => pp(aa_real_bool(Q,X4)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(P)),collect_real(Q))) ) ).

% Collect_mono
tff(fact_1256_Collect__mono,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
         => pp(aa_list_nat_bool(Q,X4)) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_1257_Collect__mono,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
         => pp(aa_set_nat_bool(Q,X4)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))) ) ).

% Collect_mono
tff(fact_1258_Collect__mono,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
         => pp(aa_nat_bool(Q,X4)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q))) ) ).

% Collect_mono
tff(fact_1259_Collect__mono,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
         => pp(aa_int_bool(Q,X4)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q))) ) ).

% Collect_mono
tff(fact_1260_subset__refl,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% subset_refl
tff(fact_1261_subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
    <=> ! [T2: nat] :
          ( pp(aa_set_nat_bool(member_nat(T2),A2))
         => pp(aa_set_nat_bool(member_nat(T2),B2)) ) ) ).

% subset_iff
tff(fact_1262_subset__iff,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
    <=> ! [T2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,T2),A2))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,T2),B2)) ) ) ).

% subset_iff
tff(fact_1263_subset__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
    <=> ! [T2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(T2),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(T2),B2)) ) ) ).

% subset_iff
tff(fact_1264_subset__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
    <=> ! [T2: real] :
          ( pp(aa_set_real_bool(member_real(T2),A2))
         => pp(aa_set_real_bool(member_real(T2),B2)) ) ) ).

% subset_iff
tff(fact_1265_subset__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ! [T2: int] :
          ( pp(aa_set_int_bool(member_int(T2),A2))
         => pp(aa_set_int_bool(member_int(T2),B2)) ) ) ).

% subset_iff
tff(fact_1266_psubset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_1267_equalityD2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ).

% equalityD2
tff(fact_1268_equalityD1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% equalityD1
tff(fact_1269_subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% subset_eq
tff(fact_1270_subset__eq,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),B2)) ) ) ).

% subset_eq
tff(fact_1271_subset__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
    <=> ! [X3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X3),B2)) ) ) ).

% subset_eq
tff(fact_1272_subset__eq,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A2))
         => pp(aa_set_real_bool(member_real(X3),B2)) ) ) ).

% subset_eq
tff(fact_1273_subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A2))
         => pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% subset_eq
tff(fact_1274_equalityE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ) ).

% equalityE
tff(fact_1275_psubsetE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ) ).

% psubsetE
tff(fact_1276_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% subsetD
tff(fact_1277_subsetD,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
       => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ) ).

% subsetD
tff(fact_1278_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% subsetD
tff(fact_1279_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
     => ( pp(aa_set_real_bool(member_real(C),A2))
       => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% subsetD
tff(fact_1280_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% subsetD
tff(fact_1281_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(member_nat(X2),A2))
       => pp(aa_set_nat_bool(member_nat(X2),B2)) ) ) ).

% in_mono
tff(fact_1282_in__mono,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2)) ) ) ).

% in_mono
tff(fact_1283_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X2: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(X2),B2)) ) ) ).

% in_mono
tff(fact_1284_in__mono,axiom,
    ! [A2: set_real,B2: set_real,X2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
     => ( pp(aa_set_real_bool(member_real(X2),A2))
       => pp(aa_set_real_bool(member_real(X2),B2)) ) ) ).

% in_mono
tff(fact_1285_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(member_int(X2),A2))
       => pp(aa_set_int_bool(member_int(X2),B2)) ) ) ).

% in_mono
tff(fact_1286_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_1287_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_1288_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_1289_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_1290_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),A4))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1291_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B4),A4))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1292_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B4),A4))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1293_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B4),A4))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1294_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),A4))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1295_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1296_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1297_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1298_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1299_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1300_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1301_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1302_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1303_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & ! [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1304_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & ! [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1305_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1306_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1307_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M4: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M4)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1308_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M4: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S2)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1309_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M4: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),M4)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1310_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M4: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1311_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_1312_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M3))
         => ? [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N5))
              & pp(aa_set_nat_bool(member_nat(N5),S2)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S2)) ) ).

% unbounded_k_infinite
tff(fact_1313_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [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(P,B5)) )
             => pp(aa_set_nat_bool(P,A5)) ) )
       => pp(aa_set_nat_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1314_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [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(P,B5)) )
             => pp(aa_set_int_bool(P,A5)) ) )
       => pp(aa_set_int_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1315_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [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(P,B5)) )
             => pp(aa_set_complex_bool(P,A5)) ) )
       => pp(aa_set_complex_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1316_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 )
       => ~ ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X)),aa_complex_real(F,lattic8794016678065449205x_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1317_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 )
       => ~ ? [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,lattic8440615504127631091l_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1318_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 )
       => ~ ? [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1319_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 )
       => ~ ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),S2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X)),aa_int_real(F,lattic2675449441010098035t_real(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1320_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 )
       => ~ ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1321_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 )
       => ~ ? [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1322_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 )
       => ~ ? [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1323_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 )
       => ~ ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),S2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1324_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 )
       => ~ ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),S2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_complex_num(F,X)),aa_complex_num(F,lattic1922116423962787043ex_num(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1325_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 )
       => ~ ? [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X)),aa_real_num(F,lattic1613168225601753569al_num(F,S2)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1326_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo6469247547558333820t_bool) ).

% bot_set_def
tff(fact_1327_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo8343409700046576086t_bool) ).

% bot_set_def
tff(fact_1328_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bo4181092155253064880l_bool) ).

% bot_set_def
tff(fact_1329_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_bool) ).

% bot_set_def
tff(fact_1330_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_bool) ).

% bot_set_def
tff(fact_1331_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1332_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M5)
     => ? [N2: nat] :
        ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X)),N2)) ) ) ).

% finite_maxlen
tff(fact_1333_finite__maxlen,axiom,
    ! [M5: set_list_bool] :
      ( finite7866827932528546246t_bool(M5)
     => ? [N2: nat] :
        ! [X: list_bool] :
          ( member_list_bool(X,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X)),N2)) ) ) ).

% finite_maxlen
tff(fact_1334_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( finite8100373058378681591st_nat(M5)
     => ? [N2: nat] :
        ! [X: list_nat] :
          ( member_list_nat(X,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X)),N2)) ) ) ).

% finite_maxlen
tff(fact_1335_finite__maxlen,axiom,
    ! [M5: set_list_int] :
      ( finite3922522038869484883st_int(M5)
     => ? [N2: nat] :
        ! [X: list_int] :
          ( member_list_int(X,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X)),N2)) ) ) ).

% finite_maxlen
tff(fact_1336_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X2: nat,M5: nat] :
      ( pp(aa_nat_bool(P,X2))
     => ( ! [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M5)) )
       => ~ ! [M3: nat] :
              ( pp(aa_nat_bool(P,M3))
             => ~ ! [X: nat] :
                    ( pp(aa_nat_bool(P,X))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),M3)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1337_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & 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),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1338_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1339_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(A),X4))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1340_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(A),A2))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & 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),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1341_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(A),A2))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & 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),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1342_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & 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),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1343_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & 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),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1344_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
            & 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),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1345_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),A))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1346_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X4: set_int] :
            ( member_set_int(X4,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(X4),A))
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1347_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(A),A2))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A2))
            & 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),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1348_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(A),A2))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A2))
            & 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),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1349_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & 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),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1350_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & 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),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1351_finite_OemptyI,axiom,
    pp(aa_set_complex_bool(finite3207457112153483333omplex,bot_bot_set_complex)) ).

% finite.emptyI
tff(fact_1352_finite_OemptyI,axiom,
    pp(aa_set_real_bool(finite_finite_real,bot_bot_set_real)) ).

% finite.emptyI
tff(fact_1353_finite_OemptyI,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,bot_bot_set_nat)) ).

% finite.emptyI
tff(fact_1354_finite_OemptyI,axiom,
    pp(aa_set_int_bool(finite_finite_int,bot_bot_set_int)) ).

% finite.emptyI
tff(fact_1355_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_1356_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_1357_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_1358_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_1359_finite__transitivity__chain,axiom,
    ! [A2: set_VEBT_VEBT,R: fun_VE5877677516038742509T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [X4: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X4),X4))
       => ( ! [X4: vEBT_VEBT,Y2: vEBT_VEBT,Z2: vEBT_VEBT] :
              ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X4),Y2))
             => ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,Y2),Z2))
               => pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X4),Z2)) ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
               => ? [Y3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),A2))
                    & pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1360_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: fun_se707272970763642173t_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( ! [X4: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),X4))
       => ( ! [X4: set_nat,Y2: set_nat,Z2: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y2))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,Y2),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),A2))
               => ? [Y3: set_nat] :
                    ( pp(aa_set_set_nat_bool(member_set_nat(Y3),A2))
                    & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1361_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: fun_co1378794544173401837x_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] : ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),X4))
       => ( ! [X4: complex,Y2: complex,Z2: complex] :
              ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y2))
             => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y2),Z2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Z2)) ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),A2))
               => ? [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),A2))
                    & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1362_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),X4))
       => ( ! [X4: real,Y2: real,Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y2),Z2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Z2)) ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),A2))
               => ? [Y3: real] :
                    ( pp(aa_set_real_bool(member_real(Y3),A2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1363_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y2: nat,Z2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y2),Z2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z2)) ) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),A2))
               => ? [Y3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y3),A2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1364_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: fun_int_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(R,X4),X4))
       => ( ! [X4: int,Y2: int,Z2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(R,Y2),Z2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Z2)) ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),A2))
               => ? [Y3: int] :
                    ( pp(aa_set_int_bool(member_int(Y3),A2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(R,X4),Y3)) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1365_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% finite_subset
tff(fact_1366_finite__subset,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% finite_subset
tff(fact_1367_finite__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% finite_subset
tff(fact_1368_infinite__super,axiom,
    ! [S2: set_nat,T3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T3)) ) ) ).

% infinite_super
tff(fact_1369_infinite__super,axiom,
    ! [S2: set_complex,T3: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T3)) ) ) ).

% infinite_super
tff(fact_1370_infinite__super,axiom,
    ! [S2: set_int,T3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,T3)) ) ) ).

% infinite_super
tff(fact_1371_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% rev_finite_subset
tff(fact_1372_rev__finite__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% rev_finite_subset
tff(fact_1373_rev__finite__subset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% rev_finite_subset
tff(fact_1374_arg__min__least,axiom,
    ! [S2: set_VEBT_VEBT,Y: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( S2 != bot_bo8194388402131092736T_VEBT )
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,lattic6139528642216935859BT_rat(F,S2))),aa_VEBT_VEBT_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1375_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_1376_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_1377_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_1378_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_1379_arg__min__least,axiom,
    ! [S2: set_VEBT_VEBT,Y: vEBT_VEBT,F: fun_VEBT_VEBT_num] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( S2 != bot_bo8194388402131092736T_VEBT )
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),S2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_VEBT_VEBT_num(F,lattic3331990488459210229BT_num(F,S2))),aa_VEBT_VEBT_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1380_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_1381_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_1382_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_1383_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_1384_subset__emptyI,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),bot_bo8194388402131092736T_VEBT)) ) ).

% subset_emptyI
tff(fact_1385_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat)) ) ).

% subset_emptyI
tff(fact_1386_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ pp(aa_set_real_bool(member_real(X4),A2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real)) ) ).

% subset_emptyI
tff(fact_1387_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ pp(aa_set_nat_bool(member_nat(X4),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat)) ) ).

% subset_emptyI
tff(fact_1388_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ pp(aa_set_int_bool(member_int(X4),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int)) ) ).

% subset_emptyI
tff(fact_1389_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_1390_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_1391_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X)) ) ).

% minf(8)
tff(fact_1392_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: 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_eq_rat,T),X)) ) ).

% minf(8)
tff(fact_1393_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X)) ) ).

% minf(8)
tff(fact_1394_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X)) ) ).

% minf(8)
tff(fact_1395_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X)) ) ).

% minf(8)
tff(fact_1396_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),T)) ) ).

% minf(6)
tff(fact_1397_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: 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_eq_rat,X),T)) ) ).

% minf(6)
tff(fact_1398_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),T)) ) ).

% minf(6)
tff(fact_1399_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),T)) ) ).

% minf(6)
tff(fact_1400_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),T)) ) ).

% minf(6)
tff(fact_1401_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X)) ) ).

% pinf(8)
tff(fact_1402_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X)) ) ).

% pinf(8)
tff(fact_1403_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X)) ) ).

% pinf(8)
tff(fact_1404_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X)) ) ).

% pinf(8)
tff(fact_1405_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X)) ) ).

% pinf(8)
tff(fact_1406_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),T)) ) ).

% pinf(6)
tff(fact_1407_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),T)) ) ).

% pinf(6)
tff(fact_1408_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),T)) ) ).

% pinf(6)
tff(fact_1409_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),T)) ) ).

% pinf(6)
tff(fact_1410_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),T)) ) ).

% pinf(6)
tff(fact_1411_complete__interval,axiom,
    ! [A: real,B: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P,A))
       => ( ~ pp(aa_real_bool(P,B))
         => ? [C3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C3),B))
              & ! [X: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C3)) )
                 => pp(aa_real_bool(P,X)) )
              & ! [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(P,X4)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1412_complete__interval,axiom,
    ! [A: nat,B: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P,A))
       => ( ~ pp(aa_nat_bool(P,B))
         => ? [C3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C3),B))
              & ! [X: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),C3)) )
                 => pp(aa_nat_bool(P,X)) )
              & ! [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(P,X4)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1413_complete__interval,axiom,
    ! [A: int,B: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P,A))
       => ( ~ pp(aa_int_bool(P,B))
         => ? [C3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C3),B))
              & ! [X: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),C3)) )
                 => pp(aa_int_bool(P,X)) )
              & ! [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(P,X4)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1414__C10_C,axiom,
    vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList),summary),xa) = aa_VEB5832953974749077948T_VEBT(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)))),aa_VEB5832953974749077948T_VEBT(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)) ).

% "10"
tff(fact_1415__C7_C,axiom,
    ( ( mi != ma )
   => ! [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))),m)))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I4 )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [Y3: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y3),na) = I4 )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y3),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),Y3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),ma)) ) ) ) ) ) ).

% "7"
tff(fact_1416_count__notin,axiom,
    ! [X2: set_nat,Xs: list_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),set_set_nat2(Xs)))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),X2) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1417_count__notin,axiom,
    ! [X2: real,Xs: list_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
     => ( aa_real_nat(count_list_real(Xs),X2) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1418_count__notin,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X2) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1419_count__notin,axiom,
    ! [X2: int,Xs: list_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
     => ( aa_int_nat(count_list_int(Xs),X2) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1420_count__notin,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
     => ( aa_nat_nat(count_list_nat(Xs),X2) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1421_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_bool,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(P,X2))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N,X2)),I))) ) ) ).

% intind
tff(fact_1422_intind,axiom,
    ! [I: nat,N: nat,P: fun_int_bool,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_int_bool(P,X2))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(replicate_int(N,X2)),I))) ) ) ).

% intind
tff(fact_1423_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBT_bool(P,X2))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X2),I))) ) ) ).

% intind
tff(fact_1424_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_1425__C2_C,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) ).

% "2"
tff(fact_1426__092_060open_062i_A_060_A2_A_094_Am_092_060close_062,axiom,
    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))) ).

% \<open>i < 2 ^ m\<close>
tff(fact_1427_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_1428_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_1429_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera1916890842035813515d_enat(M) = numera1916890842035813515d_enat(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_1430_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_1431_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera6620942414471956472nteger(M) = numera6620942414471956472nteger(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_1432_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_1433__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_1434__C4_Oprems_C,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"
tff(fact_1435__C4_OIH_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),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)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_vebt_insert(X,Xa)),na)) ) ) ) ).

% "4.IH"(1)
tff(fact_1436_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_12)) )
          & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1437__C4_C,axiom,
    ! [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))),m)))
     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I4)),X_13))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I4)) ) ) ).

% "4"
tff(fact_1438_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: 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),M))))
     => 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))),M))) ) ).

% high_bound_aux
tff(fact_1439_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X2))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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_1440_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1441_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1442_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1443_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1444_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1445_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1446_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_1447_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_1448_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_1449_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_1450_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_1451_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_1452_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_1453_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_1454_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_1455_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_1456_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_1457_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: code_integer] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(V),plus_p5714425477246183910nteger(numera6620942414471956472nteger(W2),Z)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W2)),Z) ).

% add_numeral_left
tff(fact_1458_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_1459_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_1460_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_1461_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_1462_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_1463_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),N)) ).

% numeral_plus_numeral
tff(fact_1464_insert__simp__mima,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        | ( X2 = 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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% insert_simp_mima
tff(fact_1465_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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),X2))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X2)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1466_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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,X2)),X2)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1467_max__bot,axiom,
    ! [X2: filter_nat] : ord_max_filter_nat(bot_bot_filter_nat,X2) = X2 ).

% max_bot
tff(fact_1468_max__bot,axiom,
    ! [X2: set_real] : ord_max_set_real(bot_bot_set_real,X2) = X2 ).

% max_bot
tff(fact_1469_max__bot,axiom,
    ! [X2: set_nat] : ord_max_set_nat(bot_bot_set_nat,X2) = X2 ).

% max_bot
tff(fact_1470_max__bot,axiom,
    ! [X2: set_int] : ord_max_set_int(bot_bot_set_int,X2) = X2 ).

% max_bot
tff(fact_1471_max__bot,axiom,
    ! [X2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X2) = X2 ).

% max_bot
tff(fact_1472_max__bot2,axiom,
    ! [X2: filter_nat] : ord_max_filter_nat(X2,bot_bot_filter_nat) = X2 ).

% max_bot2
tff(fact_1473_max__bot2,axiom,
    ! [X2: set_real] : ord_max_set_real(X2,bot_bot_set_real) = X2 ).

% max_bot2
tff(fact_1474_max__bot2,axiom,
    ! [X2: set_nat] : ord_max_set_nat(X2,bot_bot_set_nat) = X2 ).

% max_bot2
tff(fact_1475_max__bot2,axiom,
    ! [X2: set_int] : ord_max_set_int(X2,bot_bot_set_int) = X2 ).

% max_bot2
tff(fact_1476_max__bot2,axiom,
    ! [X2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),bot_bot_nat) = X2 ).

% max_bot2
tff(fact_1477_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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_vebt_member,vEBT_vebt_insert(T,X2)),Y))
           => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
              | ( X2 = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1478_replicate__eq__replicate,axiom,
    ! [M: nat,X2: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X2) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X2 = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1479_length__replicate,axiom,
    ! [N: nat,X2: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X2)) = N ).

% length_replicate
tff(fact_1480_length__replicate,axiom,
    ! [N: nat,X2: bool] : size_size_list_bool(replicate_bool(N,X2)) = N ).

% length_replicate
tff(fact_1481_length__replicate,axiom,
    ! [N: nat,X2: nat] : size_size_list_nat(replicate_nat(N,X2)) = N ).

% length_replicate
tff(fact_1482_length__replicate,axiom,
    ! [N: nat,X2: int] : size_size_list_int(replicate_int(N,X2)) = N ).

% length_replicate
tff(fact_1483_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_1484_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_1485_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_1486_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_1487_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_1488_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_1489_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_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_1490__C4_OIH_C_I2_J,axiom,
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_vebt_insert(summary,X2)),m)) ) ).

% "4.IH"(2)
tff(fact_1491_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_1492_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_1493_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(1)
tff(fact_1494_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_1495_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_1496_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_1497_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X2)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(4)
tff(fact_1498_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X2)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(4)
tff(fact_1499_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X2)),zero_zero_int) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(4)
tff(fact_1500_max__0__1_I4_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X2),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X2) ).

% max_0_1(4)
tff(fact_1501_max__0__1_I4_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X2)),zero_zero_real) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(4)
tff(fact_1502_max__0__1_I4_J,axiom,
    ! [X2: num] : ord_max_Code_integer(numera6620942414471956472nteger(X2),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(X2) ).

% max_0_1(4)
tff(fact_1503_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,X2)) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(3)
tff(fact_1504_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X2)) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(3)
tff(fact_1505_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,X2)) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(3)
tff(fact_1506_max__0__1_I3_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(X2)) = numera1916890842035813515d_enat(X2) ).

% max_0_1(3)
tff(fact_1507_max__0__1_I3_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,X2)) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(3)
tff(fact_1508_max__0__1_I3_J,axiom,
    ! [X2: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(X2)) = numera6620942414471956472nteger(X2) ).

% max_0_1(3)
tff(fact_1509__C6_C,axiom,
    ( 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))) ) ).

% "6"
tff(fact_1510_myIHs,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),na))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),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_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_vebt_insert(X2,Xa2)),na)) ) ) ) ).

% myIHs
tff(fact_1511_in__set__replicate,axiom,
    ! [X2: set_nat,N: nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X2),set_set_nat2(replicate_set_nat(N,Y))))
    <=> ( ( X2 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1512_in__set__replicate,axiom,
    ! [X2: real,N: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X2),set_real2(replicate_real(N,Y))))
    <=> ( ( X2 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1513_in__set__replicate,axiom,
    ! [X2: int,N: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X2),set_int2(replicate_int(N,Y))))
    <=> ( ( X2 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1514_in__set__replicate,axiom,
    ! [X2: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(replicate_nat(N,Y))))
    <=> ( ( X2 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1515_in__set__replicate,axiom,
    ! [X2: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
    <=> ( ( X2 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_1516_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ? [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N,A))))
          & pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1517_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1518_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
          & pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_1519_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N,A))))
         => pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1520_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1521_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_1522_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(nth_nat(replicate_nat(N,X2)),I) = X2 ) ) ).

% nth_replicate
tff(fact_1523_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_int(nth_int(replicate_int(N,X2)),I) = X2 ) ) ).

% nth_replicate
tff(fact_1524_nth__replicate,axiom,
    ! [I: nat,N: nat,X2: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X2),I) = X2 ) ) ).

% nth_replicate
tff(fact_1525_highlowprop,axiom,
    ( 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))),m)))
    & 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))) ) ).

% highlowprop
tff(fact_1526_add__One__commute,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),N) = aa_num_num(plus_plus_num(N),one) ).

% add_One_commute
tff(fact_1527_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_1528_le__num__One__iff,axiom,
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),one))
    <=> ( X2 = one ) ) ).

% le_num_One_iff
tff(fact_1529_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_1530_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_1531_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_1532_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_1533_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_1534_numeral__Bit0,axiom,
    ! [N: num] : numera6620942414471956472nteger(aa_num_num(bit0,N)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(N)) ).

% numeral_Bit0
tff(fact_1535_max__absorb2,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
     => ( ord_max_set_int(X2,Y) = Y ) ) ).

% max_absorb2
tff(fact_1536_max__absorb2,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
     => ( aa_rat_rat(ord_max_rat(X2),Y) = Y ) ) ).

% max_absorb2
tff(fact_1537_max__absorb2,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y))
     => ( aa_num_num(ord_max_num(X2),Y) = Y ) ) ).

% max_absorb2
tff(fact_1538_max__absorb2,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y) = Y ) ) ).

% max_absorb2
tff(fact_1539_max__absorb2,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y) = Y ) ) ).

% max_absorb2
tff(fact_1540_max__absorb1,axiom,
    ! [Y: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X2))
     => ( ord_max_set_int(X2,Y) = X2 ) ) ).

% max_absorb1
tff(fact_1541_max__absorb1,axiom,
    ! [Y: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X2))
     => ( aa_rat_rat(ord_max_rat(X2),Y) = X2 ) ) ).

% max_absorb1
tff(fact_1542_max__absorb1,axiom,
    ! [Y: num,X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X2))
     => ( aa_num_num(ord_max_num(X2),Y) = X2 ) ) ).

% max_absorb1
tff(fact_1543_max__absorb1,axiom,
    ! [Y: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y) = X2 ) ) ).

% max_absorb1
tff(fact_1544_max__absorb1,axiom,
    ! [Y: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X2))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y) = X2 ) ) ).

% max_absorb1
tff(fact_1545_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_1546_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_1547_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_1548_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_1549_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_1550_max__add__distrib__right,axiom,
    ! [X2: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),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,X2),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Z)) ).

% max_add_distrib_right
tff(fact_1551_max__add__distrib__right,axiom,
    ! [X2: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X2),Y)),aa_rat_rat(plus_plus_rat(X2),Z)) ).

% max_add_distrib_right
tff(fact_1552_max__add__distrib__right,axiom,
    ! [X2: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),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,X2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Z)) ).

% max_add_distrib_right
tff(fact_1553_max__add__distrib__right,axiom,
    ! [X2: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),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,X2),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Z)) ).

% max_add_distrib_right
tff(fact_1554_max__add__distrib__left,axiom,
    ! [X2: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_1555_max__add__distrib__left,axiom,
    ! [X2: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(X2),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X2),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ).

% max_add_distrib_left
tff(fact_1556_max__add__distrib__left,axiom,
    ! [X2: 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,X2),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_1557_max__add__distrib__left,axiom,
    ! [X2: 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,X2),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_1558_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),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,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).

% nat_add_max_right
tff(fact_1559_nat__add__max__left,axiom,
    ! [M: 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,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).

% nat_add_max_left
tff(fact_1560_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_1561_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_1562_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_1563_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_1564_zero__neq__numeral,axiom,
    ! [N: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(N) ).

% zero_neq_numeral
tff(fact_1565_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_1566_zero__neq__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != numera6620942414471956472nteger(N) ).

% zero_neq_numeral
tff(fact_1567_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1568_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_1569_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_1570_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),zero_z3403309356797280102nteger)) ).

% not_numeral_le_zero
tff(fact_1571_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_1572_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_1573_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_1574_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_le_numeral
tff(fact_1575_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_1576_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(N))) ).

% zero_le_numeral
tff(fact_1577_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_1578_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_1579_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_1580_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_1581_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_1582_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_1583_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_1584_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_1585_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),zero_z3403309356797280102nteger)) ).

% not_numeral_less_zero
tff(fact_1586_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_1587_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_1588_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_1589_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N))) ).

% zero_less_numeral
tff(fact_1590_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_1591_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(N))) ).

% zero_less_numeral
tff(fact_1592_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X2: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N )
     => ( ! [Y2: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(Y2),set_set_nat2(Xs)))
           => ( Y2 = X2 ) )
       => ( Xs = replicate_set_nat(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_1593_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X2: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y2: real] :
            ( pp(aa_set_real_bool(member_real(Y2),set_real2(Xs)))
           => ( Y2 = X2 ) )
       => ( Xs = replicate_real(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_1594_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X2: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),set_VEBT_VEBT2(Xs)))
           => ( Y2 = X2 ) )
       => ( Xs = replicate_VEBT_VEBT(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_1595_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X2: bool] :
      ( ( size_size_list_bool(Xs) = N )
     => ( ! [Y2: bool] :
            ( member_bool(Y2,set_bool2(Xs))
           => ( pp(Y2)
            <=> pp(X2) ) )
       => ( Xs = replicate_bool(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_1596_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X2: nat] :
      ( ( size_size_list_nat(Xs) = N )
     => ( ! [Y2: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y2),set_nat2(Xs)))
           => ( Y2 = X2 ) )
       => ( Xs = replicate_nat(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_1597_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X2: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y2: int] :
            ( pp(aa_set_int_bool(member_int(Y2),set_int2(Xs)))
           => ( Y2 = X2 ) )
       => ( Xs = replicate_int(N,X2) ) ) ) ).

% replicate_eqI
tff(fact_1598_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => ( X4 = X2 ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_1599_replicate__length__same,axiom,
    ! [Xs: list_bool,X2: bool] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => ( pp(X4)
          <=> pp(X2) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_1600_replicate__length__same,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( X4 = X2 ) )
     => ( replicate_nat(size_size_list_nat(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_1601_replicate__length__same,axiom,
    ! [Xs: list_int,X2: int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => ( X4 = X2 ) )
     => ( replicate_int(size_size_list_int(Xs),X2) = Xs ) ) ).

% replicate_length_same
tff(fact_1602_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_1603_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
           => ( ( pp(aa_real_bool(P,X))
                & pp(aa_real_bool(Q,X)) )
            <=> ( pp(aa_real_bool(P2,X))
                & pp(aa_real_bool(Q3,X)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1604_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: 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(P,X4))
          <=> pp(aa_rat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
           => ( ( pp(aa_rat_bool(P,X))
                & pp(aa_rat_bool(Q,X)) )
            <=> ( pp(aa_rat_bool(P2,X))
                & pp(aa_rat_bool(Q3,X)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1605_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: 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(P,X4))
          <=> pp(aa_num_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
           => ( ( pp(aa_num_bool(P,X))
                & pp(aa_num_bool(Q,X)) )
            <=> ( pp(aa_num_bool(P2,X))
                & pp(aa_num_bool(Q3,X)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1606_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: 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(P,X4))
          <=> pp(aa_nat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
           => ( ( pp(aa_nat_bool(P,X))
                & pp(aa_nat_bool(Q,X)) )
            <=> ( pp(aa_nat_bool(P2,X))
                & pp(aa_nat_bool(Q3,X)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1607_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: 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(P,X4))
          <=> pp(aa_int_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
           => ( ( pp(aa_int_bool(P,X))
                & pp(aa_int_bool(Q,X)) )
            <=> ( pp(aa_int_bool(P2,X))
                & pp(aa_int_bool(Q3,X)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1608_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
           => ( ( pp(aa_real_bool(P,X))
                | pp(aa_real_bool(Q,X)) )
            <=> ( pp(aa_real_bool(P2,X))
                | pp(aa_real_bool(Q3,X)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1609_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: 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(P,X4))
          <=> pp(aa_rat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
           => ( ( pp(aa_rat_bool(P,X))
                | pp(aa_rat_bool(Q,X)) )
            <=> ( pp(aa_rat_bool(P2,X))
                | pp(aa_rat_bool(Q3,X)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1610_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: 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(P,X4))
          <=> pp(aa_num_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
           => ( ( pp(aa_num_bool(P,X))
                | pp(aa_num_bool(Q,X)) )
            <=> ( pp(aa_num_bool(P2,X))
                | pp(aa_num_bool(Q3,X)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1611_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: 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(P,X4))
          <=> pp(aa_nat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
           => ( ( pp(aa_nat_bool(P,X))
                | pp(aa_nat_bool(Q,X)) )
            <=> ( pp(aa_nat_bool(P2,X))
                | pp(aa_nat_bool(Q3,X)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1612_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: 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(P,X4))
          <=> pp(aa_int_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
           => ( ( pp(aa_int_bool(P,X))
                | pp(aa_int_bool(Q,X)) )
            <=> ( pp(aa_int_bool(P2,X))
                | pp(aa_int_bool(Q3,X)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1613_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ( X != T ) ) ).

% pinf(3)
tff(fact_1614_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => ( X != T ) ) ).

% pinf(3)
tff(fact_1615_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
     => ( X != T ) ) ).

% pinf(3)
tff(fact_1616_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => ( X != T ) ) ).

% pinf(3)
tff(fact_1617_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => ( X != T ) ) ).

% pinf(3)
tff(fact_1618_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ( X != T ) ) ).

% pinf(4)
tff(fact_1619_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => ( X != T ) ) ).

% pinf(4)
tff(fact_1620_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
     => ( X != T ) ) ).

% pinf(4)
tff(fact_1621_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => ( X != T ) ) ).

% pinf(4)
tff(fact_1622_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => ( X != T ) ) ).

% pinf(4)
tff(fact_1623_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T)) ) ).

% pinf(5)
tff(fact_1624_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),T)) ) ).

% pinf(5)
tff(fact_1625_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),T)) ) ).

% pinf(5)
tff(fact_1626_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),T)) ) ).

% pinf(5)
tff(fact_1627_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),T)) ) ).

% pinf(5)
tff(fact_1628_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X)) ) ).

% pinf(7)
tff(fact_1629_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X)) ) ).

% pinf(7)
tff(fact_1630_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X)) ) ).

% pinf(7)
tff(fact_1631_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X)) ) ).

% pinf(7)
tff(fact_1632_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X)) ) ).

% pinf(7)
tff(fact_1633_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
           => ( ( pp(aa_real_bool(P,X))
                & pp(aa_real_bool(Q,X)) )
            <=> ( pp(aa_real_bool(P2,X))
                & pp(aa_real_bool(Q3,X)) ) ) ) ) ) ).

% minf(1)
tff(fact_1634_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: 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(P,X4))
          <=> pp(aa_rat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
           => ( ( pp(aa_rat_bool(P,X))
                & pp(aa_rat_bool(Q,X)) )
            <=> ( pp(aa_rat_bool(P2,X))
                & pp(aa_rat_bool(Q3,X)) ) ) ) ) ) ).

% minf(1)
tff(fact_1635_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: 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(P,X4))
          <=> pp(aa_num_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
           => ( ( pp(aa_num_bool(P,X))
                & pp(aa_num_bool(Q,X)) )
            <=> ( pp(aa_num_bool(P2,X))
                & pp(aa_num_bool(Q3,X)) ) ) ) ) ) ).

% minf(1)
tff(fact_1636_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: 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(P,X4))
          <=> pp(aa_nat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
           => ( ( pp(aa_nat_bool(P,X))
                & pp(aa_nat_bool(Q,X)) )
            <=> ( pp(aa_nat_bool(P2,X))
                & pp(aa_nat_bool(Q3,X)) ) ) ) ) ) ).

% minf(1)
tff(fact_1637_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: 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(P,X4))
          <=> pp(aa_int_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
           => ( ( pp(aa_int_bool(P,X))
                & pp(aa_int_bool(Q,X)) )
            <=> ( pp(aa_int_bool(P2,X))
                & pp(aa_int_bool(Q3,X)) ) ) ) ) ) ).

% minf(1)
tff(fact_1638_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
           => ( ( pp(aa_real_bool(P,X))
                | pp(aa_real_bool(Q,X)) )
            <=> ( pp(aa_real_bool(P2,X))
                | pp(aa_real_bool(Q3,X)) ) ) ) ) ) ).

% minf(2)
tff(fact_1639_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: 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(P,X4))
          <=> pp(aa_rat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
           => ( ( pp(aa_rat_bool(P,X))
                | pp(aa_rat_bool(Q,X)) )
            <=> ( pp(aa_rat_bool(P2,X))
                | pp(aa_rat_bool(Q3,X)) ) ) ) ) ) ).

% minf(2)
tff(fact_1640_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: 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(P,X4))
          <=> pp(aa_num_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
           => ( ( pp(aa_num_bool(P,X))
                | pp(aa_num_bool(Q,X)) )
            <=> ( pp(aa_num_bool(P2,X))
                | pp(aa_num_bool(Q3,X)) ) ) ) ) ) ).

% minf(2)
tff(fact_1641_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: 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(P,X4))
          <=> pp(aa_nat_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
           => ( ( pp(aa_nat_bool(P,X))
                | pp(aa_nat_bool(Q,X)) )
            <=> ( pp(aa_nat_bool(P2,X))
                | pp(aa_nat_bool(Q3,X)) ) ) ) ) ) ).

% minf(2)
tff(fact_1642_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: 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(P,X4))
          <=> pp(aa_int_bool(P2,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(Q3,X4)) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
           => ( ( pp(aa_int_bool(P,X))
                | pp(aa_int_bool(Q,X)) )
            <=> ( pp(aa_int_bool(P2,X))
                | pp(aa_int_bool(Q3,X)) ) ) ) ) ) ).

% minf(2)
tff(fact_1643_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => ( X != T ) ) ).

% minf(3)
tff(fact_1644_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
     => ( X != T ) ) ).

% minf(3)
tff(fact_1645_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
     => ( X != T ) ) ).

% minf(3)
tff(fact_1646_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
     => ( X != T ) ) ).

% minf(3)
tff(fact_1647_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
     => ( X != T ) ) ).

% minf(3)
tff(fact_1648_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => ( X != T ) ) ).

% minf(4)
tff(fact_1649_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
     => ( X != T ) ) ).

% minf(4)
tff(fact_1650_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
     => ( X != T ) ) ).

% minf(4)
tff(fact_1651_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
     => ( X != T ) ) ).

% minf(4)
tff(fact_1652_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
     => ( X != T ) ) ).

% minf(4)
tff(fact_1653_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T)) ) ).

% minf(5)
tff(fact_1654_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: 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,X),T)) ) ).

% minf(5)
tff(fact_1655_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),T)) ) ).

% minf(5)
tff(fact_1656_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),T)) ) ).

% minf(5)
tff(fact_1657_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),T)) ) ).

% minf(5)
tff(fact_1658_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X)) ) ).

% minf(7)
tff(fact_1659_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X: 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,T),X)) ) ).

% minf(7)
tff(fact_1660_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X)) ) ).

% minf(7)
tff(fact_1661_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X)) ) ).

% minf(7)
tff(fact_1662_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X)) ) ).

% minf(7)
tff(fact_1663_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X2: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X2)),size_s6755466524823107622T_VEBT(Xs))) ).

% count_le_length
tff(fact_1664_count__le__length,axiom,
    ! [Xs: list_bool,X2: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(count_list_bool(Xs),X2)),size_size_list_bool(Xs))) ).

% count_le_length
tff(fact_1665_count__le__length,axiom,
    ! [Xs: list_nat,X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),X2)),size_size_list_nat(Xs))) ).

% count_le_length
tff(fact_1666_count__le__length,axiom,
    ! [Xs: list_int,X2: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),X2)),size_size_list_int(Xs))) ).

% count_le_length
tff(fact_1667_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: 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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [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))),M)))
                   => ( ? [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_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( 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))),M)))
                             => ( ( ( 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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_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_1668_sum__power2__eq__zero__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),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 )
    <=> ( ( X2 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1669_sum__power2__eq__zero__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X2 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1670_sum__power2__eq__zero__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X2),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 )
    <=> ( ( X2 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1671_zero__less__power2,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1672_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_1673_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_1674_power2__eq__iff__nonneg,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X2 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1675_power2__eq__iff__nonneg,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_nat_rat(power_power_rat(X2),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))) )
        <=> ( X2 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1676_power2__eq__iff__nonneg,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2))
     => ( 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,X2),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))) )
        <=> ( X2 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1677_power2__eq__iff__nonneg,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X2),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))) )
        <=> ( X2 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1678_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1679_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_1680_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_1681_insert__simp__norm,axiom,
    ! [X2: 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,X2),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),X2))
       => ( 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))
         => ( ( X2 != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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,X2),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(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,X2),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,X2),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_1682_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X2: 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,X2),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))
         => ( ( X2 != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),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))))))),aa_VEB5832953974749077948T_VEBT(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_1683_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_1684_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_1685_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1686_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_1687_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_1688_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(power_power_real(A),N)),aa_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_1689_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_1690_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_1691_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_1692_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_1693_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_1694_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_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_1695_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_1696_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_1697_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2))
     => ( 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))
        & ( ( X2 = Mi )
          | ( X2 = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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,X2),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_1698_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_1699_high__def,axiom,
    ! [X2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),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_1700__C9_C,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = na ).

% "9"
tff(fact_1701_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1702_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_1703_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_1704_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1705_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_1706_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_1707_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_1708_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_1709_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1710_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1711_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_1712_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1713_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_1714_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1715_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1716_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1717_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1718_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1719_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1720_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1721_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_1722_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_1723_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_1724_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_1725_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_1726_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_1727_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_1728_nat__zero__less__power__iff,axiom,
    ! [X2: 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,X2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1729_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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,X2),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,X2),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,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),X2)) ) ) ) ).

% both_member_options_ding
tff(fact_1730_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_1731_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_1732_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_1733_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_1734_divide__right__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_right_mono
tff(fact_1735_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_right_mono
tff(fact_1736_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_1737_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_1738_divide__nonneg__nonneg,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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,X2),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_1739_divide__nonneg__nonneg,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( 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(X2),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_1740_divide__nonneg__nonpos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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,X2),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_1741_divide__nonneg__nonpos,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( 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(X2),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_1742_divide__nonpos__nonneg,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_1743_divide__nonpos__nonneg,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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(X2),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_1744_divide__nonpos__nonpos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_1745_divide__nonpos__nonpos,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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(X2),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_1746_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: 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,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_1747_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: 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,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_1748_divide__neg__neg,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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(X2),Y))) ) ) ).

% divide_neg_neg
tff(fact_1749_divide__neg__neg,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2),Y))) ) ) ).

% divide_neg_neg
tff(fact_1750_divide__neg__pos,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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(X2),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_1751_divide__neg__pos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_1752_divide__pos__neg,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ( 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(X2),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_1753_divide__pos__neg,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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,X2),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_1754_divide__pos__pos,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ( 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(X2),Y))) ) ) ).

% divide_pos_pos
tff(fact_1755_divide__pos__pos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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,X2),Y))) ) ) ).

% divide_pos_pos
tff(fact_1756_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_1757_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_1758_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_1759_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_1760_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_1761_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_1762_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_1763_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_1764_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: 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,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1765_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1766_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_1767_frac__le,axiom,
    ! [Y: real,X2: 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,X2),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,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_1768_frac__le,axiom,
    ! [Y: rat,X2: 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,X2),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(X2),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_1769_frac__less,axiom,
    ! [X2: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_1770_frac__less,axiom,
    ! [X2: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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(X2),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_1771_frac__less2,axiom,
    ! [X2: real,Y: real,W2: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_1772_frac__less2,axiom,
    ! [X2: rat,Y: rat,W2: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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(X2),Z)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_1773_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_1774_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_1775_divide__nonneg__neg,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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,X2),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_1776_divide__nonneg__neg,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( 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(X2),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_1777_divide__nonneg__pos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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,X2),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_1778_divide__nonneg__pos,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( 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(X2),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_1779_divide__nonpos__neg,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_1780_divide__nonpos__neg,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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(X2),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_1781_divide__nonpos__pos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_1782_divide__nonpos__pos,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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(X2),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_1783_field__sum__of__halves,axiom,
    ! [X2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X2 ).

% field_sum_of_halves
tff(fact_1784_field__sum__of__halves,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X2 ).

% field_sum_of_halves
tff(fact_1785_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_1786_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_1787_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_1788_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_1789_field__less__half__sum,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_1790_field__less__half__sum,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_1791_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_1792_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_1793_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),N) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_1794_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_1795_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_1796_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(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ).

% power_mono
tff(fact_1797_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_1798_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_1799_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_1800_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(power_power_real(A),N))) ) ).

% zero_le_power
tff(fact_1801_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_1802_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_1803_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_1804_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(power_power_real(A),N))) ) ).

% zero_less_power
tff(fact_1805_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_1806_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_1807_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_1808_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_1809_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1810_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(power_power_real(A),N)),aa_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_1817_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_1818_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_1819_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(power_power_real(A),N) = aa_nat_real(power_power_real(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1820_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_1821_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_1822_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_1823_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),N) = aa_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_zero__power2,axiom,
    aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_1830_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_1831_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_1832_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_1833_self__le__ge2__pow,axiom,
    ! [K: nat,M: 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))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_1834_power2__nat__le__eq__le,axiom,
    ! [M: 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,M),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,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_1835_power2__nat__le__imp__le,axiom,
    ! [M: 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,M),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,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_1836_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb)),X2))
    <=> ( ( X2 = Mi )
        | ( X2 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1837_zero__le__power2,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_1838_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_1839_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_1840_power2__eq__imp__eq,axiom,
    ! [X2: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X2 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1841_power2__eq__imp__eq,axiom,
    ! [X2: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X2),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),X2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => ( X2 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1842_power2__eq__imp__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),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),X2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X2 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1843_power2__eq__imp__eq,axiom,
    ! [X2: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X2),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),X2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X2 = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1844_power2__le__imp__le,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( 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,X2),Y)) ) ) ).

% power2_le_imp_le
tff(fact_1845_power2__le__imp__le,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X2),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,X2),Y)) ) ) ).

% power2_le_imp_le
tff(fact_1846_power2__le__imp__le,axiom,
    ! [X2: 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,X2),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,X2),Y)) ) ) ).

% power2_le_imp_le
tff(fact_1847_power2__le__imp__le,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X2),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,X2),Y)) ) ) ).

% power2_le_imp_le
tff(fact_1848_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(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_1849_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_1850_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_1851_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_1852_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_1853_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_1854_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_1855_power2__less__imp__less,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( 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,X2),Y)) ) ) ).

% power2_less_imp_less
tff(fact_1856_power2__less__imp__less,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X2),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,X2),Y)) ) ) ).

% power2_less_imp_less
tff(fact_1857_power2__less__imp__less,axiom,
    ! [X2: 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,X2),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,X2),Y)) ) ) ).

% power2_less_imp_less
tff(fact_1858_power2__less__imp__less,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X2),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,X2),Y)) ) ) ).

% power2_less_imp_less
tff(fact_1859_sum__power2__ge__zero,axiom,
    ! [X2: 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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_1860_sum__power2__ge__zero,axiom,
    ! [X2: 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(X2),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_1861_sum__power2__ge__zero,axiom,
    ! [X2: 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(X2),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_1862_sum__power2__le__zero__iff,axiom,
    ! [X2: 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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X2 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1863_sum__power2__le__zero__iff,axiom,
    ! [X2: 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(X2),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))
    <=> ( ( X2 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1864_sum__power2__le__zero__iff,axiom,
    ! [X2: 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(X2),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))
    <=> ( ( X2 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1865_not__sum__power2__lt__zero,axiom,
    ! [X2: 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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_1866_not__sum__power2__lt__zero,axiom,
    ! [X2: 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(X2),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_1867_not__sum__power2__lt__zero,axiom,
    ! [X2: 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(X2),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_1868_sum__power2__gt__zero__iff,axiom,
    ! [X2: 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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X2 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1869_sum__power2__gt__zero__iff,axiom,
    ! [X2: 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(X2),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))))))
    <=> ( ( X2 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1870_sum__power2__gt__zero__iff,axiom,
    ! [X2: 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(X2),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))))))
    <=> ( ( X2 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1871_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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),M))))
     => ( 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),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1872_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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),M))))
     => ( 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),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),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_1873_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X2: 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,X2),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,X2),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,X2),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,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_1874_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_1875_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: 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,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X2))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X2 = Mi )
          | ( X2 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_1876_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_1877_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_1878_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_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))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_1879_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N: nat] : divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_1880_div__exp__eq,axiom,
    ! [A: nat,M: 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))),M))),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,M),N))) ).

% div_exp_eq
tff(fact_1881_div__exp__eq,axiom,
    ! [A: int,M: 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))),M))),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,M),N))) ).

% div_exp_eq
tff(fact_1882_exp__add__not__zero__imp__left,axiom,
    ! [M: 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,M),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))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1883_exp__add__not__zero__imp__left,axiom,
    ! [M: 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,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1884_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1885_exp__add__not__zero__imp__right,axiom,
    ! [M: 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,M),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_1886_exp__add__not__zero__imp__right,axiom,
    ! [M: 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,M),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_1887_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1888_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_1889_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1890_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1891_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1892_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1893_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1894_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1895_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1896_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1897_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1898_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1899_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_1900_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_1901_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_1902_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_1903_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_1904_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1905_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_1906_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1907_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_1908_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_1909_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1910_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_1911_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1912_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_1913_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_1914_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_1915_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_1916_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_1917_numeral__times__numeral,axiom,
    ! [M: num,N: num] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_1918_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_1919_numeral__times__numeral,axiom,
    ! [M: num,N: num] : times_3573771949741848930nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(aa_num_num(times_times_num(M),N)) ).

% numeral_times_numeral
tff(fact_1920_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1921_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W2)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1922_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W2)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1923_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),times_7803423173614009249d_enat(numera1916890842035813515d_enat(W2),Z)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(times_times_num(V),W2)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1924_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1925_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(V),times_3573771949741848930nteger(numera6620942414471956472nteger(W2),Z)) = times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(times_times_num(V),W2)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1926_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_1927_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_1928_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_1929_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1930_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_1931_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_1932_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1933_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_1934_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_1935_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_1936_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_1937_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_1938_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_1939_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_1940_high__inv,axiom,
    ! [X2: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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))),X2)),N) = Y ) ) ).

% high_inv
tff(fact_1941_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(K),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,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_1942_low__inv,axiom,
    ! [X2: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),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))),X2)),N) = X2 ) ) ).

% low_inv
tff(fact_1943_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C) ).

% times_divide_eq_left
tff(fact_1944_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_1945_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_1946_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_1947_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),B) ).

% divide_divide_eq_right
tff(fact_1948_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_1949_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C) ).

% times_divide_eq_right
tff(fact_1950_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_1951_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_1952_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_1953_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_1954_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_1955_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_1956_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1957_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1958_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1959_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1960_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1961_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1962_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(78)
tff(fact_1963_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),one)) ).

% semiring_norm(75)
tff(fact_1964_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1965_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1966_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1967_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1968_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_1969_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1970_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1971_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1972_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1973_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1974_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1975_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1976_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_1977_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_1978_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1979_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1980_sum__squares__eq__zero__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X2 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1981_sum__squares__eq__zero__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( X2 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1982_sum__squares__eq__zero__iff,axiom,
    ! [X2: 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,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X2 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1983_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_1984_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_1985_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_1986_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_1987_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_1988_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_1989_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1990_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1991_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1992_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1993_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1994_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1995_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_1996_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_1997_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_1998_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_1999_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_2000_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2001_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2002_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2003_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2004_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2005_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2006_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_2007_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_2008_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_2009_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_2010_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_2011_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2012_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2013_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2014_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_2015_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_2016_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_2017_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_2018_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_2019_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B),numera6620942414471956472nteger(V)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(A,numera6620942414471956472nteger(V)),times_3573771949741848930nteger(B,numera6620942414471956472nteger(V))) ).

% distrib_right_numeral
tff(fact_2020_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_2021_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_2022_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_2023_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(B,C)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),B),times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),C)) ).

% distrib_left_numeral
tff(fact_2024_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_2025_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(V),plus_p5714425477246183910nteger(B,C)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V),B),times_3573771949741848930nteger(numera6620942414471956472nteger(V),C)) ).

% distrib_left_numeral
tff(fact_2026_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_2027_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_2028_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_2029_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2030_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2031_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_2032_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_2033_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_2034_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_2035_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_2036_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_2037_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2038_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2039_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_2040_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_2041_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_2042_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_2043_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_2044_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_2045_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_2046_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_2047_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_2048_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_2049_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_2050_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_2051_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2052_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_rat(numeral_numeral_rat,N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2053_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2054_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2055_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat(N) = one_on7984719198319812577d_enat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2056_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_real(numeral_numeral_real,N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2057_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger(N) = one_one_Code_integer )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2058_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2059_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2060_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2061_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2062_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_on7984719198319812577d_enat = numera1916890842035813515d_enat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2063_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2064_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer = numera6620942414471956472nteger(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2065_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(power_power_real(A),M) = aa_nat_real(power_power_real(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2066_power__inject__exp,axiom,
    ! [A: rat,M: 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),M) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2067_power__inject__exp,axiom,
    ! [A: nat,M: 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),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2068_power__inject__exp,axiom,
    ! [A: int,M: 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),M) = aa_nat_int(power_power_int(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2069_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_2070_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_2071_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_2072_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_2073_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_2074_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_2075_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_2076_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_2077_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X2)),one_one_rat) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(6)
tff(fact_2078_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X2)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(6)
tff(fact_2079_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X2)),one_one_int) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(6)
tff(fact_2080_max__0__1_I6_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X2),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X2) ).

% max_0_1(6)
tff(fact_2081_max__0__1_I6_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X2)),one_one_real) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(6)
tff(fact_2082_max__0__1_I6_J,axiom,
    ! [X2: num] : ord_max_Code_integer(numera6620942414471956472nteger(X2),one_one_Code_integer) = numera6620942414471956472nteger(X2) ).

% max_0_1(6)
tff(fact_2083_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X2)) = aa_num_rat(numeral_numeral_rat,X2) ).

% max_0_1(5)
tff(fact_2084_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X2)) = aa_num_nat(numeral_numeral_nat,X2) ).

% max_0_1(5)
tff(fact_2085_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,X2)) = aa_num_int(numeral_numeral_int,X2) ).

% max_0_1(5)
tff(fact_2086_max__0__1_I5_J,axiom,
    ! [X2: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X2)) = numera1916890842035813515d_enat(X2) ).

% max_0_1(5)
tff(fact_2087_max__0__1_I5_J,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,X2)) = aa_num_real(numeral_numeral_real,X2) ).

% max_0_1(5)
tff(fact_2088_max__0__1_I5_J,axiom,
    ! [X2: num] : ord_max_Code_integer(one_one_Code_integer,numera6620942414471956472nteger(X2)) = numera6620942414471956472nteger(X2) ).

% max_0_1(5)
tff(fact_2089_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2090_mult__less__cancel2,axiom,
    ! [M: nat,K: 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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_2091_nat__0__less__mult__iff,axiom,
    ! [M: 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,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_2092_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_2093_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
      & ( ( K != 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_2094_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_2095_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_2096_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_2097_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_2098_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_2099_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_2100_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_2101_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_2102_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_2103_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_2104_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_2105_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_2106_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_2107_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_2108_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_2109_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_2110_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_2111_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_2112_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_2113_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_2114_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_2115_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_2116_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_2117_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_2118_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_2119_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_2120_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_2121_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_2122_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_2123_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_2124_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_2125_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_2126_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_2127_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_2128_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_2129_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_2130_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_2131_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_2132_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_2133_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_2134_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_2135_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_2136_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_2137_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_2138_power__strict__increasing__iff,axiom,
    ! [B: real,X2: 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(power_power_real(B),X2)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2139_power__strict__increasing__iff,axiom,
    ! [B: rat,X2: 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),X2)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2140_power__strict__increasing__iff,axiom,
    ! [B: nat,X2: 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),X2)),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,X2),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2141_power__strict__increasing__iff,axiom,
    ! [B: int,X2: 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),X2)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2142_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2143_mult__le__cancel2,axiom,
    ! [M: nat,K: 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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_2144_div__mult__self1__is__m,axiom,
    ! [N: nat,M: 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),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_2145_div__mult__self__is__m,axiom,
    ! [N: nat,M: 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,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_2146_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_2147_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_2148_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_2149_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_2150_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_2151_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_2152_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_2153_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_2154_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_2155_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_2156_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_2157_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_2158_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_2159_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_2160_one__add__one,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2161_power__strict__decreasing__iff,axiom,
    ! [B: real,M: 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(power_power_real(B),M)),aa_nat_real(power_power_real(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2162_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: 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),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2163_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: 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),M)),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),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2164_power__strict__decreasing__iff,axiom,
    ! [B: int,M: 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),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2165_power__increasing__iff,axiom,
    ! [B: real,X2: 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(power_power_real(B),X2)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ) ).

% power_increasing_iff
tff(fact_2166_power__increasing__iff,axiom,
    ! [B: rat,X2: 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),X2)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ) ).

% power_increasing_iff
tff(fact_2167_power__increasing__iff,axiom,
    ! [B: nat,X2: 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),X2)),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,X2),Y)) ) ) ).

% power_increasing_iff
tff(fact_2168_power__increasing__iff,axiom,
    ! [B: int,X2: 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),X2)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ) ).

% power_increasing_iff
tff(fact_2169_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_2170_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_2171_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_2172_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_2173_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_2174_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_2175_numeral__plus__one,axiom,
    ! [N: num] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(N),one)) ).

% numeral_plus_one
tff(fact_2176_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_2177_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_2178_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_2179_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_2180_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_2181_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_2182_one__plus__numeral,axiom,
    ! [N: num] : plus_p5714425477246183910nteger(one_one_Code_integer,numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),N)) ).

% one_plus_numeral
tff(fact_2183_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_2184_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_2185_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),one_one_Code_integer))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2195_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_2196_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_2197_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_2198_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_2199_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_2200_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_2201_power__decreasing__iff,axiom,
    ! [B: real,M: 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(power_power_real(B),M)),aa_nat_real(power_power_real(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2202_power__decreasing__iff,axiom,
    ! [B: rat,M: 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),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2203_power__decreasing__iff,axiom,
    ! [B: nat,M: 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),M)),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),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2204_power__decreasing__iff,axiom,
    ! [B: int,M: 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),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2205_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2206_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2207_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2208_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_2209_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_2210_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_2211_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_2212_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_2213_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_2214_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_2215_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_2216_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_2217_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_2218_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_2219_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_2220_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_2221_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_2222_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2223_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_2224_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2225_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_2226_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_2227_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.left_commute
tff(fact_2228_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_2229_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_2230_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_2231_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_2232_one__reorient,axiom,
    ! [X2: complex] :
      ( ( one_one_complex = X2 )
    <=> ( X2 = one_one_complex ) ) ).

% one_reorient
tff(fact_2233_one__reorient,axiom,
    ! [X2: real] :
      ( ( one_one_real = X2 )
    <=> ( X2 = one_one_real ) ) ).

% one_reorient
tff(fact_2234_one__reorient,axiom,
    ! [X2: rat] :
      ( ( one_one_rat = X2 )
    <=> ( X2 = one_one_rat ) ) ).

% one_reorient
tff(fact_2235_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
    <=> ( X2 = one_one_nat ) ) ).

% one_reorient
tff(fact_2236_one__reorient,axiom,
    ! [X2: int] :
      ( ( one_one_int = X2 )
    <=> ( X2 = one_one_int ) ) ).

% one_reorient
tff(fact_2237_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_2238_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( 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,M),N))) ) ) ).

% less_1_mult
tff(fact_2239_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( 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(M),N))) ) ) ).

% less_1_mult
tff(fact_2240_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( 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,M),N))) ) ) ).

% less_1_mult
tff(fact_2241_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( 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,M),N))) ) ) ).

% less_1_mult
tff(fact_2242_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2243_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2244_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2245_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),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),C)),A)) ) ) ).

% mult_left_le
tff(fact_2246_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),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),C)),A)) ) ) ).

% mult_left_le
tff(fact_2247_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),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),C)),A)) ) ) ).

% mult_left_le
tff(fact_2248_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),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),C)),A)) ) ) ).

% mult_left_le
tff(fact_2249_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_2250_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_2251_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_2252_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_2253_mult__right__le__one__le,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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,X2),Y)),X2)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2254_mult__right__le__one__le,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( 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(X2),Y)),X2)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2255_mult__right__le__one__le,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( 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,X2),Y)),X2)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2256_mult__left__le__one__le,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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),X2)),X2)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2257_mult__left__le__one__le,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( 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),X2)),X2)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2258_mult__left__le__one__le,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( 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),X2)),X2)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2259_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(power_power_real(A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2260_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_2261_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_2262_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_2263_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(power_power_real(A),N)))) ) ).

% power_gt1_lemma
tff(fact_2264_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_2265_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_2266_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_2267_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2268_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2269_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2270_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2271_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2272_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2273_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2274_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2275_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2276_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2277_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_2278_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_2279_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_2280_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_2281_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_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_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_2292_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2293_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2294_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2295_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2296_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X2: 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,X2),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,X2),Y)) )
    <=> ( ( W2 = X2 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2297_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X2: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Y)),aa_rat_rat(times_times_rat(X2),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Z)),aa_rat_rat(times_times_rat(X2),Y)) )
    <=> ( ( W2 = X2 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2298_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X2: 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,X2),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,X2),Y)) )
    <=> ( ( W2 = X2 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2299_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X2: 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,X2),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,X2),Y)) )
    <=> ( ( W2 = X2 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2300_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2301_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2302_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2303_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2304_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_2305_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_2306_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_2307_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_2308_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_2309_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_2310_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_2311_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_2312_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2313_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_2314_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2315_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2316_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2317_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2318_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2319_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2320_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2321_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2322_times__divide__times__eq,axiom,
    ! [X2: rat,Y: rat,Z: rat,W2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(X2),Y)),aa_rat_rat(divide_divide_rat(Z),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X2),Z)),aa_rat_rat(times_times_rat(Y),W2)) ).

% times_divide_times_eq
tff(fact_2323_times__divide__times__eq,axiom,
    ! [X2: 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,X2),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,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W2)) ).

% times_divide_times_eq
tff(fact_2324_divide__divide__times__eq,axiom,
    ! [X2: rat,Y: rat,Z: rat,W2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(X2),Y)),aa_rat_rat(divide_divide_rat(Z),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X2),W2)),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2325_divide__divide__times__eq,axiom,
    ! [X2: 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,X2),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,X2),W2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_2326_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_2327_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_2328_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_2329_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_2330_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_2331_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_2332_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_2333_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2334_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2335_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2336_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2337_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2338_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_2339_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_2340_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_2341_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_2342_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),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,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

% mult_le_mono2
tff(fact_2343_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),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,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

% mult_le_mono1
tff(fact_2344_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_2345_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_2346_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).

% le_cube
tff(fact_2347_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% add_mult_distrib2
tff(fact_2348_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% add_mult_distrib
tff(fact_2349_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_2350_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: 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(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2351_nat__mult__max__left,axiom,
    ! [M: 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,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_2352_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),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,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_max_right
tff(fact_2353_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_2354_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: 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,M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_2355_div__times__less__eq__dividend,axiom,
    ! [M: 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,M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_2356_field__le__mult__one__interval,axiom,
    ! [X2: 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),X2)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2357_field__le__mult__one__interval,axiom,
    ! [X2: 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),X2)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2358_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2359_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => 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,C),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_2360_mult__less__cancel__right2,axiom,
    ! [A: int,C: 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),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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,C),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_2361_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2362_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => 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,C),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_2363_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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,C),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_2364_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2365_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => 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,C),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_2366_mult__less__cancel__left2,axiom,
    ! [C: 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,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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,C),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_2367_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2368_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => 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,C),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_2369_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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,C),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_2370_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2371_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),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_2372_mult__le__cancel__right2,axiom,
    ! [A: int,C: 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),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),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_2373_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2374_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),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_2375_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),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_2376_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2377_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),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_2378_mult__le__cancel__left2,axiom,
    ! [C: 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,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),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_2379_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2380_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),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_2381_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),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_2382_convex__bound__le,axiom,
    ! [X2: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2383_convex__bound__le,axiom,
    ! [X2: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),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),X2)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2384_convex__bound__le,axiom,
    ! [X2: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),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),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2385_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(power_power_real(A),N))),aa_nat_real(power_power_real(A),N))) ) ) ).

% power_Suc_less
tff(fact_2386_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_2387_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_2388_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_2389_convex__bound__lt,axiom,
    ! [X2: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2390_convex__bound__lt,axiom,
    ! [X2: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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),X2)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2391_convex__bound__lt,axiom,
    ! [X2: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),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),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2392_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2393_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2394_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2395_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2396_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_2397_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_2398_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_2399_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_2400_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_2401_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_2402_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_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_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_2420_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_2421_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_2422_mult__right__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono
tff(fact_2423_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono
tff(fact_2424_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_right_mono
tff(fact_2425_mult__right__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono
tff(fact_2426_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: 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,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2427_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: 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,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2428_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: 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,C),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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2429_mult__left__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono
tff(fact_2430_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono
tff(fact_2431_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_left_mono
tff(fact_2432_mult__left__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono
tff(fact_2433_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_2434_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_2435_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_2436_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: 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,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2437_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: 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,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2438_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: 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,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2439_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_2440_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_2441_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_2442_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_2443_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_2444_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_2445_mult__mono_H,axiom,
    ! [A: real,B: real,C: 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,C),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),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2446_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: 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,C),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),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2447_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C))
           => 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2448_mult__mono_H,axiom,
    ! [A: int,B: int,C: 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,C),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),C))
           => 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2449_mult__mono,axiom,
    ! [A: real,B: real,C: 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,C),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),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2450_mult__mono,axiom,
    ! [A: rat,B: rat,C: 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,C),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),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2451_mult__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C))
           => 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2452_mult__mono,axiom,
    ! [A: int,B: int,C: 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,C),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),C))
           => 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2453_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2454_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2455_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2456_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2457_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2458_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & 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,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2459_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & 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,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2460_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2461_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2462_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2463_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_2464_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2465_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: 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,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2466_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: 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,C),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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2467_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2468_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & 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,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2469_mult__less__cancel__left__disj,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & 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,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2470_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2471_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2472_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2473_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_2474_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2475_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: 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,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2476_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: 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,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2477_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2478_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2479_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2480_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2481_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2482_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2483_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_2484_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_2485_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_2486_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_2487_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_2488_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_2489_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_2490_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_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_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_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_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_2505_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_2506_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_2507_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_2508_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_2509_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_2510_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_2511_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_2512_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_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2520_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2521_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2522_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2523_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2524_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X2: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X2),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z) )
        <=> ( aa_complex_complex(times_times_complex(X2),Z) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2525_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X2),Y) = aa_rat_rat(divide_divide_rat(W2),Z) )
        <=> ( aa_rat_rat(times_times_rat(X2),Z) = aa_rat_rat(times_times_rat(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2526_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X2: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2527_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_2528_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_2529_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_2530_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_2531_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_2532_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_2533_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2534_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2535_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2536_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2537_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2538_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2539_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2540_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2541_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2542_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2543_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2544_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2545_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_2546_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_2547_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_2548_mult__numeral__1,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(one),A) = A ).

% mult_numeral_1
tff(fact_2549_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_2550_mult__numeral__1,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(one),A) = A ).

% mult_numeral_1
tff(fact_2551_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_2552_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_2553_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_2554_mult__numeral__1__right,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(one)) = A ).

% mult_numeral_1_right
tff(fact_2555_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_2556_mult__numeral__1__right,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(A,numera6620942414471956472nteger(one)) = A ).

% mult_numeral_1_right
tff(fact_2557_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_2558_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_2559_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_2560_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_2561_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_2562_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: 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,M),Q2)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_2563_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: 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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_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_2572_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_2573_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_2574_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_2575_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_2576_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_2577_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_2578_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_2579_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_2580_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_2581_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_2582_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_2583_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_2584_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_2585_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_2586_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_2587_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_2588_power__add,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_2589_power__add,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)) ).

% power_add
tff(fact_2590_power__add,axiom,
    ! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_2591_power__add,axiom,
    ! [A: nat,M: 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,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_2592_power__add,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_2593_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).

% mult_less_mono1
tff(fact_2594_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).

% mult_less_mono2
tff(fact_2595_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N))) ).

% one_le_numeral
tff(fact_2596_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_2597_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),numera6620942414471956472nteger(N))) ).

% one_le_numeral
tff(fact_2598_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_2599_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_2600_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_2601_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_2602_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_2603_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_2604_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_2605_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_2606_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),one_one_Code_integer)) ).

% not_numeral_less_one
tff(fact_2607_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_2608_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_2609_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_2610_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_2611_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_2612_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_2613_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_2614_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_2615_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_2616_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_2617_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_2618_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X2)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X2)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_2619_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X2)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X2)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_2620_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X2)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_2621_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X2)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_2622_one__plus__numeral__commute,axiom,
    ! [X2: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X2)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X2),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_2623_one__plus__numeral__commute,axiom,
    ! [X2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X2)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2624_one__plus__numeral__commute,axiom,
    ! [X2: num] : plus_p5714425477246183910nteger(one_one_Code_integer,numera6620942414471956472nteger(X2)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(X2),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_2625_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_2626_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_2627_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_2628_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_2629_numeral__One,axiom,
    numera1916890842035813515d_enat(one) = one_on7984719198319812577d_enat ).

% numeral_One
tff(fact_2630_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_2631_numeral__One,axiom,
    numera6620942414471956472nteger(one) = one_one_Code_integer ).

% numeral_One
tff(fact_2632_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(power_power_real(A),N))) ) ).

% one_le_power
tff(fact_2633_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_2634_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_2635_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_2636_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2637_div__less__dividend,axiom,
    ! [N: nat,M: 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),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_2638_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2639_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_2640_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2641_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2642_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2643_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_2644_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: 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,M),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,M),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2645_dividend__less__times__div,axiom,
    ! [N: nat,M: 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,M),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,M),N))))) ) ).

% dividend_less_times_div
tff(fact_2646_dividend__less__div__times,axiom,
    ! [N: nat,M: 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,M),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,M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_2647_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I5)),J3) )
               => pp(aa_nat_bool(P,I5)) ) ) ) ) ) ).

% split_div
tff(fact_2648_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: 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,C),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),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2649_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: 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,C),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),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2650_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2651_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: 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,C),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),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2652_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: 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,C),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),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2653_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: 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,C),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),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2654_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2655_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: 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,C),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),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2656_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2657_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2658_mult__right__le__imp__le,axiom,
    ! [A: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2659_mult__right__le__imp__le,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2660_mult__left__le__imp__le,axiom,
    ! [C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2661_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2662_mult__left__le__imp__le,axiom,
    ! [C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2663_mult__left__le__imp__le,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2664_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2665_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2666_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2667_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2668_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2669_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2670_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2671_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => 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,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2672_mult__less__cancel__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2673_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: 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,C),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),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2674_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: 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,C),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),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2675_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2676_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: 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,C),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),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2677_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2678_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2679_mult__right__less__imp__less,axiom,
    ! [A: nat,C: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2680_mult__right__less__imp__less,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2681_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2682_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => 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,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2683_mult__less__cancel__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => 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,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2684_mult__strict__mono,axiom,
    ! [A: real,B: real,C: 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,C),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),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2685_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: 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,C),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),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2686_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2687_mult__strict__mono,axiom,
    ! [A: int,B: int,C: 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,C),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),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2688_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2689_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2690_mult__left__less__imp__less,axiom,
    ! [C: 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,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2691_mult__left__less__imp__less,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2692_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2693_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2694_mult__le__cancel__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2695_mult__le__cancel__left,axiom,
    ! [C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2696_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2697_mult__le__cancel__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => 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,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2698_sum__squares__ge__zero,axiom,
    ! [X2: 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,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2699_sum__squares__ge__zero,axiom,
    ! [X2: 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(X2),X2)),aa_rat_rat(times_times_rat(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2700_sum__squares__ge__zero,axiom,
    ! [X2: 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,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2701_sum__squares__le__zero__iff,axiom,
    ! [X2: 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,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X2 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2702_sum__squares__le__zero__iff,axiom,
    ! [X2: 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(X2),X2)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat))
    <=> ( ( X2 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2703_sum__squares__le__zero__iff,axiom,
    ! [X2: 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,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X2 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2704_not__sum__squares__lt__zero,axiom,
    ! [X2: 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,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_2705_not__sum__squares__lt__zero,axiom,
    ! [X2: 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(X2),X2)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_2706_not__sum__squares__lt__zero,axiom,
    ! [X2: 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,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_2707_sum__squares__gt__zero__iff,axiom,
    ! [X2: 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,X2),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X2 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2708_sum__squares__gt__zero__iff,axiom,
    ! [X2: 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(X2),X2)),aa_rat_rat(times_times_rat(Y),Y))))
    <=> ( ( X2 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2709_sum__squares__gt__zero__iff,axiom,
    ! [X2: 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,X2),X2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X2 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2710_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: 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,C),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(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2711_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2712_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: 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),C))
       => ( 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(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2713_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: 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),C))
       => ( 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_2714_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X2: 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)),X2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(X2),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2715_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X2: 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)),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_2716_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X2: 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,X2),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(X2),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2717_mult__imp__div__pos__less,axiom,
    ! [Y: real,X2: 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,X2),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,X2),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_2718_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_2719_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_2720_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2721_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_2722_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2723_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_2724_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_2725_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_2726_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2727_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_2728_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2729_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_2730_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2731_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2732_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2733_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W2) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2734_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2735_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W2) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2736_divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X2),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X2),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2737_divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2738_divide__add__eq__iff,axiom,
    ! [Z: real,X2: 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,X2),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2739_add__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X2),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X2),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2740_add__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(X2),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(X2),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2741_add__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),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,X2),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2742_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X2: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X2),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X2),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2743_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X2: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(X2),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2744_add__num__frac,axiom,
    ! [Y: real,Z: real,X2: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2745_add__frac__num,axiom,
    ! [Y: complex,X2: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X2),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X2),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2746_add__frac__num,axiom,
    ! [Y: rat,X2: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X2),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2747_add__frac__num,axiom,
    ! [Y: real,X2: 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,X2),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2748_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X2: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X2),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X2),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2749_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X2),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(X2),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2750_add__frac__eq,axiom,
    ! [Y: real,Z: real,X2: 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,X2),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,X2),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_2751_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_2752_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_2753_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_2754_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_2755_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_2756_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_2757_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_2758_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_2759_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_2760_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_2761_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_2762_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_2763_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_2764_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_2765_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(power_power_real(A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2766_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_2767_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_2768_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_2769_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_2770_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_2771_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_2772_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_2773_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_2774_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_2775_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_2776_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_2777_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_2778_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(power_power_real(A),N)),aa_nat_real(power_power_real(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_2779_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_2780_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_2781_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_2782_power__less__imp__less__exp,axiom,
    ! [A: real,M: 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(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2783_power__less__imp__less__exp,axiom,
    ! [A: rat,M: 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),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2784_power__less__imp__less__exp,axiom,
    ! [A: nat,M: 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),M)),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,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2785_power__less__imp__less__exp,axiom,
    ! [A: int,M: 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),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_2786_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(power_power_real(A),N)),aa_nat_real(power_power_real(A),N4))) ) ) ).

% power_increasing
tff(fact_2787_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_2788_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_2789_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_2790_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: 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,C),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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2791_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: 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,C),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(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_2792_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X2: 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)),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2793_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X2: 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)),X2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(X2),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_2794_mult__imp__div__pos__le,axiom,
    ! [Y: real,X2: 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,X2),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,X2),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2795_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X2: 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,X2),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(X2),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_2796_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_2797_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_2798_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_2799_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_2800_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_2801_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_2802_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_2803_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_2804_divide__left__mono,axiom,
    ! [B: real,A: real,C: 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),C))
       => ( 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono
tff(fact_2805_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: 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),C))
       => ( 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(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono
tff(fact_2806_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_2807_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_2808_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_2809_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_2810_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_2811_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_2812_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: 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),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_2813_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: 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),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_2814_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_2815_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_2816_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_2817_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_2818_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_2819_mult__2,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Z) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2
tff(fact_2820_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_2821_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_2822_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_2823_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_2824_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_2825_mult__2__right,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(Z,numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2_right
tff(fact_2826_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_2827_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_2828_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_2829_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_2830_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_2831_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A),B) ).

% left_add_twice
tff(fact_2832_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_2833_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_2834_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_2835_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_2836_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(power_power_real(A),N4)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2837_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_2838_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_2839_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_2840_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(power_power_real(A),N4)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2841_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_2842_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_2843_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_2844_power__le__imp__le__exp,axiom,
    ! [A: real,M: 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(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2845_power__le__imp__le__exp,axiom,
    ! [A: rat,M: 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),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2846_power__le__imp__le__exp,axiom,
    ! [A: nat,M: 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),M)),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,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2847_power__le__imp__le__exp,axiom,
    ! [A: int,M: 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),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2848_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(power_power_real(A),N))) ) ) ).

% self_le_power
tff(fact_2849_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_2850_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_2851_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_2852_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(power_power_real(A),N))) ) ) ).

% one_less_power
tff(fact_2853_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_2854_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_2855_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_2856_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_2857_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: 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),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_2858_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: 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),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_2859_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_2860_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_2861_power2__sum,axiom,
    ! [X2: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X2),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(X2),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))),X2)),Y)) ).

% power2_sum
tff(fact_2862_power2__sum,axiom,
    ! [X2: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(X2),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(X2),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))),X2)),Y)) ).

% power2_sum
tff(fact_2863_power2__sum,axiom,
    ! [X2: 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,X2),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,X2),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))),X2)),Y)) ).

% power2_sum
tff(fact_2864_power2__sum,axiom,
    ! [X2: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),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(X2),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))),X2)),Y)) ).

% power2_sum
tff(fact_2865_power2__sum,axiom,
    ! [X2: extended_enat,Y: extended_enat] : power_8040749407984259932d_enat(plus_p3455044024723400733d_enat(X2,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(power_8040749407984259932d_enat(X2,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)),X2),Y)) ).

% power2_sum
tff(fact_2866_power2__sum,axiom,
    ! [X2: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(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))),X2)),Y)) ).

% power2_sum
tff(fact_2867_power2__sum,axiom,
    ! [X2: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(X2,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_3573771949741848930nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),X2),Y)) ).

% power2_sum
tff(fact_2868_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(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_2869_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_2870_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_2871_div__le__dividend,axiom,
    ! [M: 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,M),N)),M)) ).

% div_le_dividend
tff(fact_2872_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).

% div_le_mono
tff(fact_2873_ex__power__ivl1,axiom,
    ! [B: nat,K: 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),K))
       => ? [N2: 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),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_2874_ex__power__ivl2,axiom,
    ! [B: nat,K: 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))),K))
       => ? [N2: 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),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_2875_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2876_div__greater__zero__iff,axiom,
    ! [M: 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,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_2877_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_2878_arith__geo__mean,axiom,
    ! [U: real,X2: real,Y: real] :
      ( ( aa_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,X2),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
       => ( 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,X2),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_2879_arith__geo__mean,axiom,
    ! [U: rat,X2: 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(X2),Y) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
       => ( 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(X2),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_2880_sum__squares__bound,axiom,
    ! [X2: 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))),X2)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_2881_sum__squares__bound,axiom,
    ! [X2: 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))),X2)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X2),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_2882_nat__induct2,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_2883_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(one_one_Code_integer,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_2884_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_2885_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_2886_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: 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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [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))),M)))
                   => ( ? [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_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( 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))),M)))
                             => ( ( ( 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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_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_2887_max__less__iff__conj,axiom,
    ! [X2: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y)),Z))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_2888_max__less__iff__conj,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(X2),Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_2889_max__less__iff__conj,axiom,
    ! [X2: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(X2),Y)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_2890_max__less__iff__conj,axiom,
    ! [X2: 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,X2),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_2891_max__less__iff__conj,axiom,
    ! [X2: 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,X2),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_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_2899_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_2900_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_2901_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_2902_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),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,C),A)) ) ) ).

% max.bounded_iff
tff(fact_2903_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),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,C),A)) ) ) ).

% max.bounded_iff
tff(fact_2904_max_Obounded__iff,axiom,
    ! [B: nat,C: 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),C)),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,C),A)) ) ) ).

% max.bounded_iff
tff(fact_2905_max_Obounded__iff,axiom,
    ! [B: int,C: 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),C)),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,C),A)) ) ) ).

% max.bounded_iff
tff(fact_2906_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_2907_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_2908_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_2909_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_2910_even__odd__cases,axiom,
    ! [X2: nat] :
      ( ! [N2: nat] : X2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : X2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_2911_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2),S3) ) ).

% deg_SUcn_Node
tff(fact_2912_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_2913_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_2914_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_2915_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_2916_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_2917_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_eq
tff(fact_2918_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% Suc_le_mono
tff(fact_2919_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_2920_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_2921_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_2922_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_2923_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_2924_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: 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),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_2925_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_2926_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_2927_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_2928_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_2929_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_2930_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_2931_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_2932_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2933_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_2934_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_2935_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_2936_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_2937_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2938_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2939_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_2940_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_2941_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_2942_nat__power__eq__Suc__0__iff,axiom,
    ! [X2: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2943_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2944_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2945_num__double,axiom,
    ! [N: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_2946_one__le__mult__iff,axiom,
    ! [M: 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,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & 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_2947_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N),one)) ).

% Suc_numeral
tff(fact_2948_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_2949_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_2950_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_2951_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,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),K))
         => ! [X: int] :
              ( pp(aa_int_bool(P,X))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_2952_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_2953_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_2954_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_2955_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_2956_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_2957_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2958_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_2959_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_2960_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_2961_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_2962_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_2963_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_2964_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_2965_Suc__inject,axiom,
    ! [X2: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y) )
     => ( X2 = Y ) ) ).

% Suc_inject
tff(fact_2966_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_2967_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(power_power_real(R3),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_2968_vebt__buildup_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( X2 != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : X2 != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_2969_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_2970_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_2971_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_2972_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_2973_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_2974_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),zero_zero_nat))
     => ( ! [Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y2)))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_2975_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_2976_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2977_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_2978_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2979_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_2980_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_2981_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_2982_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_lessD
tff(fact_2983_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_2984_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ( aa_nat_nat(suc,M) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_2985_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_2986_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_2987_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I5)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
            & pp(aa_nat_bool(P,I5)) ) ) ) ).

% Ex_less_Suc
tff(fact_2988_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_2989_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_2990_All__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I5)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
           => pp(aa_nat_bool(P,I5)) ) ) ) ).

% All_less_Suc
tff(fact_2991_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
    <=> ? [M6: nat] :
          ( ( M = aa_nat_nat(suc,M6) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M6)) ) ) ).

% Suc_less_eq2
tff(fact_2992_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_2993_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_2994_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_2995_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(suc,I3)))
       => ( ! [I3: nat,J2: nat,K3: 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),K3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K3))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),K3)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_2996_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => pp(aa_nat_bool(P,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I3)))
               => pp(aa_nat_bool(P,I3)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_2997_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_2998_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_leD
tff(fact_2999_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_3000_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_3001_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))
     => ? [M3: nat] : M7 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_3002_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_3003_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_3004_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_3005_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_3006_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_3007_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y2: nat,Z2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y2),Z2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z2)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_3008_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_3009_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_3010_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_3011_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_3012_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( 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_3013_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( 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_3014_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( 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_3015_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( 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_3016_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( 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_3017_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3018_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3019_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3020_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3021_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3022_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2))))
     => ( 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_3023_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( 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_3024_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( 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_3025_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( 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_3026_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( 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_3027_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2)))
     => ( 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_3028_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
     => ( 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_3029_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
     => ( 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_3030_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
     => ( 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_3031_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N6: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
     => ( 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_3032_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = 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_3033_gr0__implies__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_implies_Suc
tff(fact_3034_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I5)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I5))) ) ) ) ).

% All_less_Suc2
tff(fact_3035_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% gr0_conv_Suc
tff(fact_3036_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I5)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I5))) ) ) ) ).

% Ex_less_Suc2
tff(fact_3037_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).

% Suc_leI
tff(fact_3038_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_eq
tff(fact_3039_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_3040_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
                 => pp(aa_nat_bool(P,N2)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_3041_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_3042_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_3043_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_3044_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_3045_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_3046_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_3047_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_3048_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q4: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ) ).

% less_natE
tff(fact_3049_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

% less_add_Suc1
tff(fact_3050_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

% less_add_Suc2
tff(fact_3051_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_iff_Suc_add
tff(fact_3052_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_imp_Suc_add
tff(fact_3053_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: 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,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_3054_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3055_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: 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,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_3056_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_3057_Suc__div__le__mono,axiom,
    ! [M: 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,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_3058_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_3059_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_3060_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_3061_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_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_3062_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_3063_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_3064_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_3065_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(power_power_real(A),aa_nat_nat(suc,N))),aa_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_3066_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_3067_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_3068_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_3069_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(power_power_real(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3070_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_3071_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_3072_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_3073_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_3074_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),K3))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K3))) ) ) ) ).

% ex_least_nat_less
tff(fact_3075_one__less__mult,axiom,
    ! [N: nat,M: 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)),M))
       => 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,M),N))) ) ) ).

% one_less_mult
tff(fact_3076_n__less__m__mult__n,axiom,
    ! [N: nat,M: 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)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_3077_n__less__n__mult__m,axiom,
    ! [N: nat,M: 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)),M))
       => 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),M))) ) ) ).

% n_less_n_mult_m
tff(fact_3078_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_3079_power__gt__expt,axiom,
    ! [N: nat,K: 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,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).

% power_gt_expt
tff(fact_3080_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => 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,I),N))) ) ).

% nat_one_le_power
tff(fact_3081_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(power_power_real(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_3082_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_3083_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_3084_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_3085_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(power_power_real(A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_3086_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_3087_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_3088_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_3089_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_3090_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: 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)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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,M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_3091_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_3092_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_3093_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_3094_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_3095_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,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)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q5))))
            & pp(aa_nat_bool(P,Q5)) ) ) ) ).

% split_div'
tff(fact_3096_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_3097_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_3098_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_3099_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(power_power_real(R3),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_3100_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(power_power_real(X4),N) = A )
            & ! [Y3: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
                  & ( aa_nat_real(power_power_real(Y3),N) = A ) )
               => ( Y3 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3101_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),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(C),D)),aa_rat_rat(ord_max_rat(A),B))) ) ) ).

% max.mono
tff(fact_3102_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),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(C),D)),aa_num_num(ord_max_num(A),B))) ) ) ).

% max.mono
tff(fact_3103_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),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,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).

% max.mono
tff(fact_3104_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),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,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ) ).

% max.mono
tff(fact_3105_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_3106_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_3107_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_3108_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_3109_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_3110_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_3111_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_3112_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_3113_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),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,C),A)) ) ) ).

% max.boundedE
tff(fact_3114_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),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,C),A)) ) ) ).

% max.boundedE
tff(fact_3115_max_OboundedE,axiom,
    ! [B: nat,C: 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),C)),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,C),A)) ) ) ).

% max.boundedE
tff(fact_3116_max_OboundedE,axiom,
    ! [B: int,C: 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),C)),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,C),A)) ) ) ).

% max.boundedE
tff(fact_3117_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: 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,C),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3118_max_OboundedI,axiom,
    ! [B: num,A: num,C: 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,C),A))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_3119_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: 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,C),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),C)),A)) ) ) ).

% max.boundedI
tff(fact_3120_max_OboundedI,axiom,
    ! [B: int,A: int,C: 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,C),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),C)),A)) ) ) ).

% max.boundedI
tff(fact_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_le__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(X2),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X2))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3134_le__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),aa_num_num(ord_max_num(X2),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X2))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3135_le__max__iff__disj,axiom,
    ! [Z: nat,X2: 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,X2),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3136_le__max__iff__disj,axiom,
    ! [Z: int,X2: 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,X2),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_3137_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_3138_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_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI1
tff(fact_3146_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.coboundedI1
tff(fact_3147_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI1
tff(fact_3148_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI1
tff(fact_3149_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI2
tff(fact_3150_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.coboundedI2
tff(fact_3151_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI2
tff(fact_3152_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI2
tff(fact_3153_less__max__iff__disj,axiom,
    ! [Z: real,X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3154_less__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(ord_max_rat(X2),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3155_less__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(ord_max_num(X2),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3156_less__max__iff__disj,axiom,
    ! [Z: nat,X2: 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,X2),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3157_less__max__iff__disj,axiom,
    ! [Z: int,X2: 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,X2),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_3158_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3159_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),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,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3160_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(B),C)),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,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3161_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: 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),C)),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,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3162_max_Ostrict__boundedE,axiom,
    ! [B: int,C: 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),C)),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,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_3163_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_3164_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_3165_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_3166_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_3167_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_3168_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3169_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3170_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3171_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3172_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_3173_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3174_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3175_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3176_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3177_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_3178_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(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_3179_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_3180_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_3181_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(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_3182_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_3183_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_3184_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_3185_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X2)) ).

% vebt_member.simps(4)
tff(fact_3186_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_3187_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_3188_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3189_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_3190_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_3191_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),X2)) ).

% vebt_member.simps(3)
tff(fact_3192_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S) ).

% vebt_insert.simps(3)
tff(fact_3193_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3194_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3195_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_3196_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_3197_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_3198_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_3199_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_3200_not__real__square__gt__zero,axiom,
    ! [X2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)))
    <=> ( X2 = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3201_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: 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),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_3202_real__arch__pow__inv,axiom,
    ! [Y: real,X2: 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,X2),one_one_real))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X2),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_3203_real__arch__pow,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(power_power_real(X2),N2))) ) ).

% real_arch_pow
tff(fact_3204_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_3205_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_3206_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_3207_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,N)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_3208_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_3209_enat__less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M2),N2))
             => pp(aa_Ext2772979874139974381t_bool(P,M2)) )
         => pp(aa_Ext2772979874139974381t_bool(P,N2)) )
     => pp(aa_Ext2772979874139974381t_bool(P,N)) ) ).

% enat_less_induct
tff(fact_3210_less__eq__real__def,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
        | ( X2 = Y ) ) ) ).

% less_eq_real_def
tff(fact_3211_q__pos__lemma,axiom,
    ! [B4: 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,B4),Q6)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_3212_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B4: 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,B4),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,B4),Q6)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B4))
         => ( 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),B4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3213_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B4: 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,B4),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,B4),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),B4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3214_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_3215_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_3216_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(divide_divide_int(N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I5: 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),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
             => pp(aa_int_bool(P,I5)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I5: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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,K),I5)),J3) ) )
             => pp(aa_int_bool(P,I5)) ) ) ) ) ).

% split_zdiv
tff(fact_3217_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A4))
     => ( 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(A4),B))) ) ) ).

% zdiv_mono1
tff(fact_3218_zdiv__mono2,axiom,
    ! [A: int,B4: 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),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),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),B4))) ) ) ) ).

% zdiv_mono2
tff(fact_3219_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(divide_divide_int(I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3220_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_3221_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_3222_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A4))
     => ( 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(A4),B)),aa_int_int(divide_divide_int(A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_3223_zdiv__mono2__neg,axiom,
    ! [A: int,B4: 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),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B4)),aa_int_int(divide_divide_int(A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3224_div__int__pos__iff,axiom,
    ! [K: 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(K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & 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,K),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_3225_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( 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(K),L))) ) ) ).

% div_positive_int
tff(fact_3226_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_3227_int__div__less__self,axiom,
    ! [X2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(X2),K)),X2)) ) ) ).

% int_div_less_self
tff(fact_3228_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_3229_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_3230_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_3231_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_3232_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3233_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_3234_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_3235_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_3236_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S) ).

% vebt_insert.simps(2)
tff(fact_3237_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,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_3238_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_3239_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_3240_mult__le__cancel__iff2,axiom,
    ! [Z: real,X2: 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),X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3241_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X2: 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),X2)),aa_rat_rat(times_times_rat(Z),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3242_mult__le__cancel__iff2,axiom,
    ! [Z: int,X2: 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),X2)),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,X2),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_3243_mult__le__cancel__iff1,axiom,
    ! [Z: real,X2: 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,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3244_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X2: 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(X2),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3245_mult__le__cancel__iff1,axiom,
    ! [Z: int,X2: 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,X2),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,X2),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_3246_divides__aux__eq,axiom,
    ! [Q2: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q2,R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_3247_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_3248_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q2,R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_3249_low__def,axiom,
    ! [X2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),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_3250_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: 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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_3251_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3252_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3253_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3254_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3255_dbl__simps_I3_J,axiom,
    neg_nu8804712462038260780nteger(one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3256_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_3257_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
    <=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).

% deg1Leaf
tff(fact_3258_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
     => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ).

% deg_1_Leaf
tff(fact_3259_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ) ).

% deg_1_Leafy
tff(fact_3260_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_3261_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_3262_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_3263_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_3264_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_3265_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_3266_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_3267_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_3268_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_3269_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_3270_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_3271_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_3272_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_3273_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_3274_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_3275_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_3276_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_3277_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_3278_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_3279_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_3280_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_3281_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_3282_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_3283_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_3284_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_3285_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_3286_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_3287_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_3288_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_3289_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_3290_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_3291_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_3292_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_3293_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_3294_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_3295_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_3296_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_3297_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_3298_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_3299_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_3300_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K)) = numera6620942414471956472nteger(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_3301_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: 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),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_3302_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_3303_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_3304_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: 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,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_3305_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3306_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_3307_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_3308_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3309_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_3310_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_3311_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_3312_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_3313_mod2__gr__0,axiom,
    ! [M: 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,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_3314_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_3315_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_3316_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
     => ( ( X2 != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3317_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_3318_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_3319_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_3320_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A4),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B4),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_3321_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_3322_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_3323_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_3324_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_3325_mod__less__eq__dividend,axiom,
    ! [M: 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,M),N)),M)) ).

% mod_less_eq_dividend
tff(fact_3326_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : X2 != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_3327_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,vEBT_Leaf(Uu,Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_3328_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_3329_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_3330_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_3331_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X2 != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3332_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X2 != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3333_vebt__insert_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X2 != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3),X4)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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_3334_vebt__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X2 != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),X4)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] : X2 != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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_3335_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
      <=> pp(Y) )
     => ( ( ( X2 = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X2 = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X2 = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3336_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_3337_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_3338_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_3339_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_3340_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_3341_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_3342_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D4: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D4)) ) ).

% mod_eqE
tff(fact_3343_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_3344_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_3345_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).

% mod_Suc
tff(fact_3346_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P3: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P3))
         => ( ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P3))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P3))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_3347_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),zero_zero_nat))
     => ( ! [M3: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_3348_mod__less__divisor,axiom,
    ! [N: nat,M: 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,M),N)),N)) ) ).

% mod_less_divisor
tff(fact_3349_mod__Suc__le__divisor,axiom,
    ! [M: 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,M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_3350_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q4: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q4) ) ).

% mod_eq_0D
tff(fact_3351_nat__mod__eq__iff,axiom,
    ! [X2: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),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,X2),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_3352_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_3353_dbl__def,axiom,
    ! [X2: real] : aa_real_real(neg_numeral_dbl_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),X2) ).

% dbl_def
tff(fact_3354_dbl__def,axiom,
    ! [X2: rat] : aa_rat_rat(neg_numeral_dbl_rat,X2) = aa_rat_rat(plus_plus_rat(X2),X2) ).

% dbl_def
tff(fact_3355_dbl__def,axiom,
    ! [X2: int] : aa_int_int(neg_numeral_dbl_int,X2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),X2) ).

% dbl_def
tff(fact_3356_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,vEBT_Leaf(Uu,Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_3357_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X2)) ).

% vebt_member.simps(2)
tff(fact_3358_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_3359_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_3360_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_3361_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_3362_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_3363_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3364_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_3365_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_3366_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_3367_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_3368_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_3369_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_3370_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_3371_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_3372_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_3373_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_3374_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_3375_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_3376_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_3377_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_3378_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_3379_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3380_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3381_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3382_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3383_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_3384_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_3385_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_3386_mod__le__divisor,axiom,
    ! [N: nat,M: 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,M),N)),N)) ) ).

% mod_le_divisor
tff(fact_3387_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( 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,A2),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),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,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ) ) ).

% div_less_mono
tff(fact_3388_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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),M))
       => ~ ! [S3: nat] : M != 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_3389_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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,M),N))
       => ~ ! [S3: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_3390_nat__mod__eq__lemma,axiom,
    ! [X2: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),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),X2))
       => ? [Q4: nat] : X2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3391_vebt__insert_Osimps_I1_J,axiom,
    ! [X2: nat,A: bool,B: bool] :
      ( ( ( X2 = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X2) = vEBT_Leaf(fTrue,B) ) )
      & ( ( X2 != zero_zero_nat )
       => ( ( ( X2 = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X2) = vEBT_Leaf(A,fTrue) ) )
          & ( ( X2 != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X2) = vEBT_Leaf(A,B) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_3392_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_3393_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,vEBT_Leaf(A,B)),X2))
    <=> ( ( ( X2 = zero_zero_nat )
         => pp(A) )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => pp(B) )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_3394_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N)) ).

% div_mod_decomp
tff(fact_3395_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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,M),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_3396_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,vEBT_Leaf(A,B)),X2))
    <=> ( ( ( X2 = zero_zero_nat )
         => pp(A) )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => pp(B) )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_3397_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X2))
     => ( ! [Uv2: bool] : X2 != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X2 != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3398_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_3399_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I5)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_3400_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3401_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3402_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( 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,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3403_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary),X2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),X2)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary) ).

% vebt_insert.simps(4)
tff(fact_3404_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B))),B))
       => ( modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3405_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_3406_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_3407_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_3408_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_3409_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_3410_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_3411_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: 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))),M)) = 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),M)))),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_3412_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: 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))),M)) = 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),M)))),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_3413_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( 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,A2),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,B2),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ).

% verit_le_mono_div
tff(fact_3414_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B))),B))
       => ( times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3415_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_3416_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_3417_mod__double__modulus,axiom,
    ! [M: code_integer,X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2))
       => ( ( modulo364778990260209775nteger(X2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) = modulo364778990260209775nteger(X2,M) )
          | ( modulo364778990260209775nteger(X2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X2,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3418_mod__double__modulus,axiom,
    ! [M: nat,X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3419_mod__double__modulus,axiom,
    ! [M: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
       => ( ( aa_int_int(modulo_modulo_int(X2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(modulo_modulo_int(X2),M) )
          | ( aa_int_int(modulo_modulo_int(X2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(X2),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3420_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_se8260200283734997820nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3421_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_3422_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_3423_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_se2793503036327961859nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3424_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_3425_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_3426_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B))))
         => ( plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3427_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_3428_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_3429_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N2)) )
                 => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                   => ( ( 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))),M3) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                         => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                           => ~ ! [X: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N2)) )
                   => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                     => ( ( 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))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                           => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                             => ~ ! [X: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N2)) )
                     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                       => ( ( 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))),M3) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                             => ( ! [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))),M3)))
                                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_12)) ) )
                                 => ( 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 )
                                         => ! [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))),M3)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N2) = I4 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N2)) )
                       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                         => ( ( 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))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                               => ( ! [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))),M3)))
                                     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_12)) ) )
                                   => ( 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 )
                                           => ! [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))),M3)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N2) = I4 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3430_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_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,N3: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N3)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N3))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N3)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N3)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N3)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N3))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [I5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I5)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I5)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),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 )
             => ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N3) = I5 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I5 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I5)),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,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N3)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N3)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [I5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I5)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I5)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),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 )
             => ! [I5: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N3) = I5 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I5 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I5)),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,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3431_mult__less__iff1,axiom,
    ! [Z: real,X2: 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,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ).

% mult_less_iff1
tff(fact_3432_mult__less__iff1,axiom,
    ! [Z: rat,X2: 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(X2),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y)) ) ) ).

% mult_less_iff1
tff(fact_3433_mult__less__iff1,axiom,
    ! [Z: int,X2: 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,X2),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,X2),Y)) ) ) ).

% mult_less_iff1
tff(fact_3434_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: 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(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_3435_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_se1345352211410354436nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3436_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_3437_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_3438_product__nth,axiom,
    ! [N: nat,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( 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_s3445333598471063425nteger(Xs)),size_s3445333598471063425nteger(Ys))))
     => ( nth_Pr2304437835452373666nteger(produc8792966785426426881nteger(Xs,Ys),N) = produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s3445333598471063425nteger(Ys))),aa_nat_Code_integer(nth_Code_integer(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s3445333598471063425nteger(Ys)))) ) ) ).

% product_nth
tff(fact_3439_product__nth,axiom,
    ! [N: nat,Xs: list_Code_integer,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_s3445333598471063425nteger(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr136253548056570280r_bool(produc2469108709554126195r_bool(Xs,Ys),N) = produc5135875973608410785r_bool(aa_nat_Code_integer(nth_Code_integer(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_3440_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_3441_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_3442_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_3443_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_3444_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_3445_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_3446_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_3447_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_3448_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3449_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3450_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3451_option_Osize_I4_J,axiom,
    ! [X22: num] : size_size_option_num(some_num(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3452_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3453_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_3454_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_3455_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3456_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_3457_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_3458_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_ri6519982836138164636nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3459_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_3460_vebt__insert_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X2,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = 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] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(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] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(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 != aa_VEB5832953974749077948T_VEBT(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)))),aa_VEB5832953974749077948T_VEBT(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))))))),aa_VEB5832953974749077948T_VEBT(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))),aa_VEB5832953974749077948T_VEBT(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_3461_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_3462_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T)) ).

% set_vebt'_def
tff(fact_3463_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3464_finite__Collect__disjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_af(P),Q))))
    <=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        & pp(aa_set_real_bool(finite_finite_real,collect_real(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3465_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_ag(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3466_finite__Collect__disjI,axiom,
    ! [P: 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(P),Q))))
    <=> ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3467_finite__Collect__disjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ai(P),Q))))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        & pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3468_finite__Collect__disjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_aj(P),Q))))
    <=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        & pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3469_finite__Collect__disjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_ak(P),Q))))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_3470_finite__Collect__conjI,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ( pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
        | 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_al(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3471_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_am(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_3472_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        | 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_an(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3473_finite__Collect__conjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        | 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_ao(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3474_finite__Collect__conjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        | 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_ap(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3475_finite__Collect__conjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        | pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_aq(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_3476_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_ar(A),B)))) ).

% finite_interval_int1
tff(fact_3477_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_as(A),B)))) ).

% finite_interval_int4
tff(fact_3478_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_3479_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_3480_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_3481_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_3482_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_3483_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_3484_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_3485_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_3486_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_3487_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_3488_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_3489_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_3490_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_3491_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_3492_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_3493_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_3494_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_3495_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_3496_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_3497_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_3498_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => ( 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,C),A)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_3499_div__dvd__div,axiom,
    ! [A: int,B: int,C: 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),C))
       => ( 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(C),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_3500_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3501_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: 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),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_3502_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_3503_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_at,A2)))) ) ).

% finite_Collect_subsets
tff(fact_3504_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_au(A2))) ) ).

% finite_Collect_subsets
tff(fact_3505_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_av(A2))) ) ).

% finite_Collect_subsets
tff(fact_3506_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_aw(A),B)))) ).

% finite_interval_int3
tff(fact_3507_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_ax(A),B)))) ).

% finite_interval_int2
tff(fact_3508_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,K)))) ).

% finite_Collect_less_nat
tff(fact_3509_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_az,K)))) ).

% finite_Collect_le_nat
tff(fact_3510_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3511_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3512_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3513_dvd__mult__cancel__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_3514_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3515_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3516_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3517_dvd__mult__cancel__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_3518_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: 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),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3519_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: 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),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3520_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: 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,C),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3521_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: 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,C),A)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3522_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_3523_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_3524_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: 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,C),A))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3525_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: 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(C),A))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3526_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: 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,C),A))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3527_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: 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,C),A))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3528_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: 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,C),A)),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3529_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: 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(C),A)),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3530_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: 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,C),A)),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3531_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: 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,C),A)),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3532_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_3533_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_3534_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_3535_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_3536_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_3537_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_3538_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_3539_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_3540_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_3541_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_3542_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_3543_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_add
tff(fact_3544_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_3545_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_3546_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_3547_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_3548_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_3549_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_3550_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_3551_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_3552_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_3553_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_3554_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_3555_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_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_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_3564_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_3565_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_3566_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_3567_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_3568_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_3569_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
    <=> ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_3570_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(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_3571_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_3572_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_3573_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_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_3574_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_3575_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_3576_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_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_3577_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_3578_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_3579_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_3580_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_3581_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer)))
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_3582_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(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_3583_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_3584_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_3585_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3586_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_3587_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_3588_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3589_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_3590_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_3591_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3592_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_3593_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_3594_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_3595_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_3596_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(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_3597_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3598_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_3599_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_3600_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_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_3601_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_3602_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_3603_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ba,I)))) ) ).

% finite_divisors_int
tff(fact_3604_Collect__restrict,axiom,
    ! [X6: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3605_Collect__restrict,axiom,
    ! [X6: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3606_Collect__restrict,axiom,
    ! [X6: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bd(X6),P)),X6) ).

% Collect_restrict
tff(fact_3607_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_be(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3608_Collect__restrict,axiom,
    ! [X6: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bf(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3609_Collect__restrict,axiom,
    ! [X6: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(X6),P))),X6)) ).

% Collect_restrict
tff(fact_3610_prop__restrict,axiom,
    ! [X2: vEBT_VEBT,Z4: set_VEBT_VEBT,X6: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),Z4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(Z4),collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(X6),P))))
       => pp(aa_VEBT_VEBT_bool(P,X2)) ) ) ).

% prop_restrict
tff(fact_3611_prop__restrict,axiom,
    ! [X2: real,Z4: set_real,X6: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(X2),Z4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(Z4),collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(X6),P))))
       => pp(aa_real_bool(P,X2)) ) ) ).

% prop_restrict
tff(fact_3612_prop__restrict,axiom,
    ! [X2: list_nat,Z4: set_list_nat,X6: set_list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(X2,Z4)
     => ( ord_le6045566169113846134st_nat(Z4,collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bd(X6),P)))
       => pp(aa_list_nat_bool(P,X2)) ) ) ).

% prop_restrict
tff(fact_3613_prop__restrict,axiom,
    ! [X2: set_nat,Z4: set_set_nat,X6: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X2),Z4))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(Z4),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_be(X6),P))))
       => pp(aa_set_nat_bool(P,X2)) ) ) ).

% prop_restrict
tff(fact_3614_prop__restrict,axiom,
    ! [X2: nat,Z4: set_nat,X6: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(X2),Z4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z4),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bf(X6),P))))
       => pp(aa_nat_bool(P,X2)) ) ) ).

% prop_restrict
tff(fact_3615_prop__restrict,axiom,
    ! [X2: int,Z4: set_int,X6: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(X2),Z4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z4),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(X6),P))))
       => pp(aa_int_bool(P,X2)) ) ) ).

% prop_restrict
tff(fact_3616_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_bh,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bh,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% subset_divisors_dvd
tff(fact_3617_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_bi,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bi,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_3618_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_ba,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_ba,B))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_3619_Collect__subset,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(A2),P))),A2)) ).

% Collect_subset
tff(fact_3620_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(A2),P))),A2)) ).

% Collect_subset
tff(fact_3621_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bd(A2),P)),A2) ).

% Collect_subset
tff(fact_3622_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_be(A2),P))),A2)) ).

% Collect_subset
tff(fact_3623_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bf(A2),P))),A2)) ).

% Collect_subset
tff(fact_3624_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(A2),P))),A2)) ).

% Collect_subset
tff(fact_3625_less__eq__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
    <=> ord_le8840051076831633728t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_eq_set_def
tff(fact_3626_less__eq__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
    <=> ord_le7893757273149057542T_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_eq_set_def
tff(fact_3627_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2))
    <=> ord_le1461730252015488778t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_eq_set_def
tff(fact_3628_less__eq__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
    <=> ord_le5573776112191778788l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_eq_set_def
tff(fact_3629_less__eq__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
    <=> ord_le5715760110363009508t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_eq_set_def
tff(fact_3630_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_3631_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_3632_dvd__trans,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_trans
tff(fact_3633_dvd__trans,axiom,
    ! [A: int,B: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_trans
tff(fact_3634_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_bj) ).

% empty_def
tff(fact_3635_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_bk) ).

% empty_def
tff(fact_3636_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_bl) ).

% empty_def
tff(fact_3637_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bm) ).

% empty_def
tff(fact_3638_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bn) ).

% empty_def
tff(fact_3639_not__finite__existsD,axiom,
    ! [P: fun_real_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,collect_real(P)))
     => ? [X_1: real] : pp(aa_real_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3640_not__finite__existsD,axiom,
    ! [P: fun_list_nat_bool] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P))
     => ? [X_1: list_nat] : pp(aa_list_nat_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3641_not__finite__existsD,axiom,
    ! [P: fun_set_nat_bool] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
     => ? [X_1: set_nat] : pp(aa_set_nat_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3642_not__finite__existsD,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ? [X_1: nat] : pp(aa_nat_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3643_not__finite__existsD,axiom,
    ! [P: fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3644_not__finite__existsD,axiom,
    ! [P: fun_complex_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
     => ? [X_1: complex] : pp(aa_complex_bool(P,X_1)) ) ).

% not_finite_existsD
tff(fact_3645_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,R: fun_VE3763956218935105199t_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_bo(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3646_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_nat,R: fun_re9121486324101951821t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bp(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3647_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_int,R: fun_VE3303673267288458579t_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B2))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_bq(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3648_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_int,R: fun_re8661203372455305201t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_br(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3649_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_complex,R: fun_VE2128610149277289041x_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B2))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_bs(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3650_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_complex,R: fun_re3010880487650781423x_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B2))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bt(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3651_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_nat,R: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_bu(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3652_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_int,R: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),B2))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R,X4),Xa)) ) )
         => ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bv(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3653_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_complex,R: fun_na5941084329111710283x_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => ? [Xa: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa),B2))
                  & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(R,X4),Xa)) ) )
         => ? [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),B2))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bw(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3654_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_nat,R: fun_int_fun_nat_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),B2))
                  & pp(aa_nat_bool(aa_int_fun_nat_bool(R,X4),Xa)) ) )
         => ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),B2))
              & ~ pp(aa_set_int_bool(finite_finite_int,collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bx(A2),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3655_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2))
    <=> ord_le6841235830588351564t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_set_def
tff(fact_3656_less__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),B2))
    <=> ord_le763421453198481914T_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_3657_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ord_le3716944864119727344t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_3658_less__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2))
    <=> ord_le4523237222650602006t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_3659_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
    <=> ord_le7519368973566061808l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_3660_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_bh,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bh,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_3661_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_bi,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bi,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_3662_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_ba,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_ba,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_3663_lambda__zero,axiom,
    aTP_Lamm_by = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_3664_lambda__zero,axiom,
    aTP_Lamm_bz = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_3665_lambda__zero,axiom,
    aTP_Lamm_ca = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_3666_lambda__zero,axiom,
    aTP_Lamm_cb = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3667_lambda__zero,axiom,
    aTP_Lamm_cc = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3668_lambda__one,axiom,
    aTP_Lamm_cd = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_3669_lambda__one,axiom,
    aTP_Lamm_ce = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_3670_lambda__one,axiom,
    aTP_Lamm_cf = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_3671_lambda__one,axiom,
    aTP_Lamm_cg = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_3672_lambda__one,axiom,
    aTP_Lamm_ch = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_3673_max__def__raw,axiom,
    ! [X: set_int,Xa: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Xa))
       => ( ord_max_set_int(X,Xa) = Xa ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(X),Xa))
       => ( ord_max_set_int(X,Xa) = X ) ) ) ).

% max_def_raw
tff(fact_3674_max__def__raw,axiom,
    ! [X: rat,Xa: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Xa))
       => ( aa_rat_rat(ord_max_rat(X),Xa) = Xa ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Xa))
       => ( aa_rat_rat(ord_max_rat(X),Xa) = X ) ) ) ).

% max_def_raw
tff(fact_3675_max__def__raw,axiom,
    ! [X: num,Xa: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Xa))
       => ( aa_num_num(ord_max_num(X),Xa) = Xa ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Xa))
       => ( aa_num_num(ord_max_num(X),Xa) = X ) ) ) ).

% max_def_raw
tff(fact_3676_max__def__raw,axiom,
    ! [X: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = X ) ) ) ).

% max_def_raw
tff(fact_3677_max__def__raw,axiom,
    ! [X: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa) = X ) ) ) ).

% max_def_raw
tff(fact_3678_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bi,M)))) ) ).

% finite_divisors_nat
tff(fact_3679_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ci(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_3680_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cj(F),U)))) ) ).

% finite_less_ub
tff(fact_3681_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_3682_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_3683_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_3684_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_3685_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_3686_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_3687_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_3688_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_3689_dvdE,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) ) ).

% dvdE
tff(fact_3690_dvdE,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) ) ).

% dvdE
tff(fact_3691_dvdE,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) ) ).

% dvdE
tff(fact_3692_dvdE,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) ) ).

% dvdE
tff(fact_3693_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).

% dvdI
tff(fact_3694_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).

% dvdI
tff(fact_3695_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% dvdI
tff(fact_3696_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% dvdI
tff(fact_3697_dvd__def,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
    <=> ? [K2: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvd_def
tff(fact_3698_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
    <=> ? [K2: rat] : A = aa_rat_rat(times_times_rat(B),K2) ) ).

% dvd_def
tff(fact_3699_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
    <=> ? [K2: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvd_def
tff(fact_3700_dvd__def,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
    <=> ? [K2: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvd_def
tff(fact_3701_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ~ ! [X4: nat,Y2: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Y2) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X4),A))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y2),B)) ) ) ) ).

% dvd_productE
tff(fact_3702_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P3),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ~ ! [X4: int,Y2: int] :
            ( ( P3 = aa_int_int(aa_int_fun_int_int(times_times_int,X4),Y2) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X4),A))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y2),B)) ) ) ) ).

% dvd_productE
tff(fact_3703_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult
tff(fact_3704_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult
tff(fact_3705_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => 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),C))) ) ).

% dvd_mult
tff(fact_3706_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => 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),C))) ) ).

% dvd_mult
tff(fact_3707_division__decomp,axiom,
    ! [A: nat,B: nat,C: 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),C)))
     => ? [B7: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C5) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B7),B))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C5),C)) ) ) ).

% division_decomp
tff(fact_3708_division__decomp,axiom,
    ! [A: int,B: int,C: 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),C)))
     => ? [B7: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B7),C5) )
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B7),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C5),C)) ) ) ).

% division_decomp
tff(fact_3709_dvd__mult2,axiom,
    ! [A: real,B: real,C: 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),C))) ) ).

% dvd_mult2
tff(fact_3710_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: 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),C))) ) ).

% dvd_mult2
tff(fact_3711_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: 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),C))) ) ).

% dvd_mult2
tff(fact_3712_dvd__mult2,axiom,
    ! [A: int,B: int,C: 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),C))) ) ).

% dvd_mult2
tff(fact_3713_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ).

% dvd_mult_left
tff(fact_3714_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ).

% dvd_mult_left
tff(fact_3715_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: 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)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ).

% dvd_mult_left
tff(fact_3716_dvd__mult__left,axiom,
    ! [A: int,B: int,C: 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)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ).

% dvd_mult_left
tff(fact_3717_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_3718_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_3719_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_3720_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_3721_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: 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,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3722_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: 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,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3723_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3724_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: 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,C),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),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_3725_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C)) ) ).

% dvd_mult_right
tff(fact_3726_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C)) ) ).

% dvd_mult_right
tff(fact_3727_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: 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)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ).

% dvd_mult_right
tff(fact_3728_dvd__mult__right,axiom,
    ! [A: int,B: int,C: 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)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ).

% dvd_mult_right
tff(fact_3729_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_3730_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_3731_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_3732_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_3733_one__dvd,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,one_one_complex),A)) ).

% one_dvd
tff(fact_3734_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_3735_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_3736_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_3737_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_3738_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_3739_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_3740_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_3741_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_3742_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: 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),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3743_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: 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),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3744_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: 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),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3745_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: 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),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_3746_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3747_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3748_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( 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),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3749_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => ( 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),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_3750_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ) ).

% dvd_add
tff(fact_3751_dvd__add,axiom,
    ! [A: rat,B: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))) ) ) ).

% dvd_add
tff(fact_3752_dvd__add,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => 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),C))) ) ) ).

% dvd_add
tff(fact_3753_dvd__add,axiom,
    ! [A: int,B: int,C: 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),C))
       => 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),C))) ) ) ).

% dvd_add
tff(fact_3754_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3755_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3756_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3757_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3758_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3759_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3760_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_3761_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_3762_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_3763_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_3764_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_3765_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_3766_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_3767_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),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,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_3768_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_3769_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),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,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3770_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3771_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_3772_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_3773_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_3774_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_3775_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_3776_numeral__code_I2_J,axiom,
    ! [N: num] : numera6620942414471956472nteger(aa_num_num(bit0,N)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(N)) ).

% numeral_code(2)
tff(fact_3777_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_3778_mod__int__pos__iff,axiom,
    ! [K: 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(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_3779_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_3780_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_3781_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S))) ) ) ).

% pinf(9)
tff(fact_3782_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S))) ) ) ).

% pinf(9)
tff(fact_3783_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),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,X),S))) ) ) ).

% pinf(9)
tff(fact_3784_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),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,X),S))) ) ) ).

% pinf(9)
tff(fact_3785_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S))) ) ) ).

% pinf(10)
tff(fact_3786_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S))) ) ) ).

% pinf(10)
tff(fact_3787_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),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,X),S))) ) ) ).

% pinf(10)
tff(fact_3788_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),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,X),S))) ) ) ).

% pinf(10)
tff(fact_3789_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S))) ) ) ).

% minf(9)
tff(fact_3790_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S))) ) ) ).

% minf(9)
tff(fact_3791_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,X),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,X),S))) ) ) ).

% minf(9)
tff(fact_3792_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),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,X),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,X),S))) ) ) ).

% minf(9)
tff(fact_3793_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S))) ) ) ).

% minf(10)
tff(fact_3794_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S))) ) ) ).

% minf(10)
tff(fact_3795_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),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,X),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,X),S))) ) ) ).

% minf(10)
tff(fact_3796_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),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,X),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,X),S))) ) ) ).

% minf(10)
tff(fact_3797_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_3798_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_3799_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_3800_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_3801_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_3802_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_3803_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_3804_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: 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,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_3805_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: 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,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_3806_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: 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)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_3807_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: 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)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_3808_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: 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),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3809_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: 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),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3810_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: 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)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3811_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: 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)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3812_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: 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),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3813_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: 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),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3814_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: 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,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3815_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: 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,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3816_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3817_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),C)),A) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3818_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3819_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3820_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),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),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3821_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),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),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3822_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: 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),C)),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3823_dvd__div__mult2__eq,axiom,
    ! [B: int,C: 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),C)),A))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3824_dvd__mult__imp__div,axiom,
    ! [A: nat,C: 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),C)),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),C))) ) ).

% dvd_mult_imp_div
tff(fact_3825_dvd__mult__imp__div,axiom,
    ! [A: int,C: 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),C)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))) ) ).

% dvd_mult_imp_div
tff(fact_3826_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: 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),C))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3827_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: 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),C))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(C),D)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3828_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: 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,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3829_unit__div__cancel,axiom,
    ! [A: int,B: int,C: 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(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3830_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: 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)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_3831_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: 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)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_3832_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: 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,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_3833_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: 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(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_3834_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3835_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3836_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3837_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3838_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_3839_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_3840_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_3841_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_3842_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_3843_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_3844_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% le_imp_power_dvd
tff(fact_3845_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3846_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3847_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N))) ) ).

% le_imp_power_dvd
tff(fact_3848_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M: 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,M),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),M)),B)) ) ) ).

% power_le_dvd
tff(fact_3849_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_3850_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M: 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,M),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_3851_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M: 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,M),N))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_3852_dvd__power__le,axiom,
    ! [X2: nat,Y: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M))) ) ) ).

% dvd_power_le
tff(fact_3853_dvd__power__le,axiom,
    ! [X2: real,Y: real,N: nat,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(X2),N)),aa_nat_real(power_power_real(Y),M))) ) ) ).

% dvd_power_le
tff(fact_3854_dvd__power__le,axiom,
    ! [X2: int,Y: int,N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X2),N)),aa_nat_int(power_power_int(Y),M))) ) ) ).

% dvd_power_le
tff(fact_3855_dvd__power__le,axiom,
    ! [X2: complex,Y: complex,N: nat,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X2),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X2),N)),aa_nat_complex(power_power_complex(Y),M))) ) ) ).

% dvd_power_le
tff(fact_3856_neg__mod__bound,axiom,
    ! [L: int,K: 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(K),L))) ) ).

% neg_mod_bound
tff(fact_3857_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: 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(K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_3858_dvd__pos__nat,axiom,
    ! [N: nat,M: 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,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_3859_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_3860_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( 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,M),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3861_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_3862_zdvd__mult__cancel,axiom,
    ! [K: int,M: 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,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N)))
     => ( ( K != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_3863_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X2: 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),X2) = 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),X2) = 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,Y2: 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)),Y2)),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),Y2)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3864_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X4: nat,Y2: 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),Y2)),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),Y2)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_3865_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,K),M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).

% zdvd_reduce
tff(fact_3866_zdvd__period,axiom,
    ! [A: int,D: int,X2: int,T: int,C: 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,X2),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,X2),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T))) ) ) ).

% zdvd_period
tff(fact_3867_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ck(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3868_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cl(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3869_finite__lists__length__eq,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cm(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3870_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cn(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3871_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_co(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_3872_even__flip__bit__iff,axiom,
    ! [M: 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,M),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))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3873_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M,A)))
    <=> ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3874_even__flip__bit__iff,axiom,
    ! [M: 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,M),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))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3875_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))) = aa_VEB5832953974749077948T_VEBT(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))) = aa_VEB5832953974749077948T_VEBT(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_3876_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_cp(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3877_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cq(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3878_finite__lists__length__le,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cr(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3879_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cs(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3880_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ct(A2),N))) ) ).

% finite_lists_length_le
tff(fact_3881_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 )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_3882_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 )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_3883_unity__coeff__ex,axiom,
    ! [P: fun_complex_bool,L: complex] :
      ( ? [X3: complex] : pp(aa_complex_bool(P,aa_complex_complex(times_times_complex(L),X3)))
    <=> ? [X3: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X3),zero_zero_complex)))
          & pp(aa_complex_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3884_unity__coeff__ex,axiom,
    ! [P: fun_real_bool,L: real] :
      ( ? [X3: real] : pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X3)))
    <=> ? [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),zero_zero_real)))
          & pp(aa_real_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3885_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X3: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X3)))
    <=> ? [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X3),zero_zero_rat)))
          & pp(aa_rat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3886_unity__coeff__ex,axiom,
    ! [P: fun_nat_bool,L: nat] :
      ( ? [X3: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3)))
    <=> ? [X3: 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,X3),zero_zero_nat)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3887_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X3: int] : pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3)))
    <=> ? [X3: 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,X3),zero_zero_int)))
          & pp(aa_int_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3888_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != 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,C),D))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3889_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != 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,C),D))
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3890_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),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),C)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3891_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3892_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: 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)),C))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3893_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: 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)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3894_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: 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) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3895_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: 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) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3896_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_3897_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_3898_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: 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,C),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3899_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: 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,C),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3900_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3901_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3902_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: 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)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3903_unit__div__commute,axiom,
    ! [B: int,A: int,C: 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)),C) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3904_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),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),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3905_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),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),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3906_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: 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,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3907_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( A = aa_int_int(divide_divide_int(C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3908_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: 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) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3909_unit__eq__div1,axiom,
    ! [B: int,A: int,C: 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) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3910_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_3911_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_3912_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_3913_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_3914_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_3915_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_3916_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_3917_neg__mod__sign,axiom,
    ! [L: int,K: 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(K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_3918_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: 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(K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3919_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),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,K),N)) ) ) ).

% dvd_imp_le
tff(fact_3920_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => ( ( aa_int_int(modulo_modulo_int(A2),N) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A2),N)),aa_int_int(divide_divide_int(B2),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_3921_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3922_dvd__mult__cancel,axiom,
    ! [K: nat,M: 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_3923_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X4: nat,Y2: 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),Y2)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3924_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_3925_div__mod__decomp__int,axiom,
    ! [A2: int,N: int] : A2 = 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(A2),N)),N)),aa_int_int(modulo_modulo_int(A2),N)) ).

% div_mod_decomp_int
tff(fact_3926_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: 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,M),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3927_vebt__buildup_Oelims,axiom,
    ! [X2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X2) = Y )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X2 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va2: nat] :
                ( ( X2 = 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 = aa_VEB5832953974749077948T_VEBT(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 = aa_VEB5832953974749077948T_VEBT(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_3928_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_3929_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_3930_even__zero,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger)) ).

% even_zero
tff(fact_3931_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_3932_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_3933_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_3934_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_3935_is__unitE,axiom,
    ! [A: nat,C: 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,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3936_is__unitE,axiom,
    ! [A: int,C: 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(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3937_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_3938_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_3939_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ) ).

% odd_even_add
tff(fact_3940_dvd__power__iff,axiom,
    ! [X2: nat,M: nat,N: nat] :
      ( ( X2 != 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,X2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_3941_dvd__power__iff,axiom,
    ! [X2: int,M: nat,N: nat] :
      ( ( X2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X2),M)),aa_nat_int(power_power_int(X2),N)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_3942_dvd__power,axiom,
    ! [N: nat,X2: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_rat ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),aa_nat_rat(power_power_rat(X2),N))) ) ).

% dvd_power
tff(fact_3943_dvd__power,axiom,
    ! [N: nat,X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X2),N))) ) ).

% dvd_power
tff(fact_3944_dvd__power,axiom,
    ! [N: nat,X2: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),aa_nat_real(power_power_real(X2),N))) ) ).

% dvd_power
tff(fact_3945_dvd__power,axiom,
    ! [N: nat,X2: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),aa_nat_int(power_power_int(X2),N))) ) ).

% dvd_power
tff(fact_3946_dvd__power,axiom,
    ! [N: nat,X2: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X2 = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X2),aa_nat_complex(power_power_complex(X2),N))) ) ).

% dvd_power
tff(fact_3947_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: 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),K)),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_3948_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3949_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3950_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3951_power__dvd__imp__le,axiom,
    ! [I: nat,M: 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,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_3952_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3953_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_3954_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_3955_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(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ).

% power_mono_odd
tff(fact_3956_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_3957_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_3958_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_3959_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_3960_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: 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),K)),K))
    <=> 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)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3961_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(modulo_modulo_int(N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I5: 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),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I5: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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,K),I5)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_3962_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_3963_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_3964_dvd__power__iff__le,axiom,
    ! [K: nat,M: 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))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_3965_even__unset__bit__iff,axiom,
    ! [M: 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,M),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))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3966_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3967_even__unset__bit__iff,axiom,
    ! [M: 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,M),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))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3968_even__set__bit__iff,axiom,
    ! [M: 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,M),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))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3969_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3970_even__set__bit__iff,axiom,
    ! [M: 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,M),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))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3971_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_3972_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList),S)),X2))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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,X2),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,X2),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_3973_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_3974_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_3975_oddE,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_3976_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_3977_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_3978_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_3979_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ) ).

% mod2_eq_if
tff(fact_3980_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_3981_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_3982_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(power_power_real(A),N))) ) ).

% zero_le_even_power
tff(fact_3983_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_3984_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_3985_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(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_3986_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_3987_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_3988_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(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_3989_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_3990_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_3991_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( 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(A2),N)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(B2),N)),zero_zero_int),one_one_int),zero_zero_int))),aa_int_int(divide_divide_int(B2),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_3992_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I5: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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,K),I5)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I5),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_3993_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I5: 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),K))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I5),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_3994_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList),Vd2)),X2))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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,X2),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,X2),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_3995_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(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)),X2))
    <=> ( ( X2 != Mi )
       => ( ( X2 != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X2))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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,X2),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,X2),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_3996_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList),Vc)),X2))
    <=> ( ( X2 = Mi )
        | ( X2 = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),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,X2),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,X2),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_3997_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(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_3998_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_3999_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_4000_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = 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] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4001_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = 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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4002_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = 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] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4003_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa2))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4004_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_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_4005_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_4006_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_4007_vebt__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = 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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4008_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa2))
     => ( ! [Uu2: bool,Uv2: bool] : X2 != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4009_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4010_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(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),X2) = aa_VEB5832953974749077948T_VEBT(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,X2),Mi)),Mi),X2)),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,X2),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Ma)))),aa_VEB5832953974749077948T_VEBT(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,X2),Mi)),X2),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,X2),Mi)),Mi),X2)),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,X2),Mi)),Mi),X2)),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,X2),Mi)),Mi),X2)),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,X2),Mi)),Mi),X2)),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_VEB5832953974749077948T_VEBT(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,X2),Mi)),Mi),X2)),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,X2),Mi)),Mi),X2)),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))),aa_VEB5832953974749077948T_VEBT(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_4011_vebt__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = 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] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X2 != aa_VEB5832953974749077948T_VEBT(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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4012_vebt__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X2 = 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] : X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X2 = aa_VEB5832953974749077948T_VEBT(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] : X2 = aa_VEB5832953974749077948T_VEBT(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] : X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4013_finite__nth__roots,axiom,
    ! [N: nat,C: 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_cu(N),C)))) ) ).

% finite_nth_roots
tff(fact_4014_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(aTP_Lamm_cv(N)))) ) ).

% finite_roots_unity
tff(fact_4015_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_cw,N)))) ) ).

% finite_roots_unity
tff(fact_4016_vebt__insert_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X2,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S3),Xa2)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S3),Xa2)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(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 = aa_VEB5832953974749077948T_VEBT(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)))),aa_VEB5832953974749077948T_VEBT(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))))))),aa_VEB5832953974749077948T_VEBT(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))),aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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_4017_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VE5569070535936029429omplex,Y: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_cx(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_cx(I6),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_cy(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4018_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_cz(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_cz(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_da(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4019_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_db(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_db(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_dc(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4020_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dd(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dd(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_de(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4021_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_df(I6),X2))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_df(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_dg(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4022_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_dh(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_dh(I6),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_di(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4023_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dj(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dj(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_dk(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4024_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dl(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dl(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_dm(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4025_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_dn(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_dn(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_do(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4026_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dp(I6),X2))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dp(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_dq(I6),X2),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_4027_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VE5569070535936029429omplex,Y: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_dr(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_dr(I6),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_ds(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4028_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dt(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dt(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_du(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4029_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_dv(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_dv(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_dw(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4030_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dx(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dx(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_dy(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4031_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dz(I6),X2))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dz(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ea(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4032_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_eb(I6),X2))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_eb(I6),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_ec(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4033_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_ed(I6),X2))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_ed(I6),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ee(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4034_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ef(I6),X2))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ef(I6),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_eg(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4035_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_eh(I6),X2))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_eh(I6),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ei(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4036_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ej(I6),X2))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ej(I6),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_ek(I6),X2),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_4037_vebt__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4038_vebt__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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(aa_VEB5832953974749077948T_VEBT(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_4039_vebt__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4040_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_4041_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
          & ( B != A ) ) ) ).

% gcd_nat.asym
tff(fact_4042_gcd__nat_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% gcd_nat.refl
tff(fact_4043_gcd__nat_Otrans,axiom,
    ! [A: nat,B: nat,C: 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),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% gcd_nat.trans
tff(fact_4044_gcd__nat_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = 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)) ) ) ).

% gcd_nat.eq_iff
tff(fact_4045_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A))
        & ( A != A ) ) ).

% gcd_nat.irrefl
tff(fact_4046_gcd__nat_Oantisym,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),A))
       => ( A = B ) ) ) ).

% gcd_nat.antisym
tff(fact_4047_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
          & ( B != C ) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_4048_gcd__nat_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: 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),C))
          & ( B != C ) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_4049_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_4050_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != 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)) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_4051_gcd__nat_Oorder__iff__strict,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,A),B))
          & ( A != B ) )
        | ( A = B ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_4052_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_4053_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% gcd_nat.strict_implies_order
tff(fact_4054_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_4055_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != 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,A),B))
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_4056_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4057_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4058_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4059_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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(aa_VEB5832953974749077948T_VEBT(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] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4060_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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] :
                    ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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] :
                      ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4061_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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] :
                  ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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_4062_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4063_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_4064_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4065_vebt__buildup_Opelims,axiom,
    ! [X2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X2) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X2))
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X2 = 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] :
                  ( ( X2 = 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 = aa_VEB5832953974749077948T_VEBT(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 = aa_VEB5832953974749077948T_VEBT(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_4066_option_Osize__gen_I2_J,axiom,
    ! [X2: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(X2,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X2,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4067_option_Osize__gen_I2_J,axiom,
    ! [X2: fun_num_nat,X22: num] : size_option_num(X2,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X2,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4068_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4069_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,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_4070_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_4071_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4072_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: 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))),M))),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),M))
        | ( 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,M),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),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4073_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: 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))),M))),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),M))
        | ( 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,M),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),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4074_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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,K),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),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_4075_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4076_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)) = 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_4077_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_4078_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ord_le8840051076831633728t_bool(aTP_Lamm_ad(R),aTP_Lamm_ad(S2))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,R),S2)) ) ).

% pred_subset_eq
tff(fact_4079_pred__subset__eq,axiom,
    ! [R: set_VEBT_VEBT,S2: set_VEBT_VEBT] :
      ( ord_le7893757273149057542T_bool(aTP_Lamm_a(R),aTP_Lamm_a(S2))
    <=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(R),S2)) ) ).

% pred_subset_eq
tff(fact_4080_pred__subset__eq,axiom,
    ! [R: set_set_nat,S2: set_set_nat] :
      ( ord_le1461730252015488778t_bool(aTP_Lamm_ac(R),aTP_Lamm_ac(S2))
    <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(R),S2)) ) ).

% pred_subset_eq
tff(fact_4081_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ord_le5573776112191778788l_bool(aTP_Lamm_aa(R),aTP_Lamm_aa(S2))
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(R),S2)) ) ).

% pred_subset_eq
tff(fact_4082_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ord_le5715760110363009508t_bool(aTP_Lamm_ae(R),aTP_Lamm_ae(S2))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(R),S2)) ) ).

% pred_subset_eq
tff(fact_4083_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4084_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4085_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4086_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4087_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_4088_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_4089_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_4090_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4091_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_4092_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_4093_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_4094_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4095_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_4096_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_4097_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_4098_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_4099_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_4100_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_4101_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_4102_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_4103_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_4104_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_4105_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_4106_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_4107_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_4108_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_4109_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_4110_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_4111_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_4112_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_4113_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_4114_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_4115_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% neg_le_iff_le
tff(fact_4116_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_4117_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_4118_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4119_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4120_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4121_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4122_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4123_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4124_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4125_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4126_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_4127_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_4128_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_4129_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_4130_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_4131_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_4132_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_4133_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_4134_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_4135_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4136_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4137_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4138_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4139_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4140_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4141_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_4142_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_4143_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_4144_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% neg_less_iff_less
tff(fact_4145_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4146_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4147_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4148_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4149_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_4150_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_4151_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_4152_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_4153_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_4154_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_4155_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_4156_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_4157_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_4158_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_4159_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_4160_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_4161_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_4162_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_4163_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_4164_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_4165_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_4166_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_4167_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_4168_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_4169_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_4170_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_4171_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_4172_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_4173_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_4174_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,times_3573771949741848930nteger(A,B)) ).

% mult_minus_right
tff(fact_4175_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_4176_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_4177_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_4178_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = times_3573771949741848930nteger(A,B) ).

% minus_mult_minus
tff(fact_4179_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_4180_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_4181_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_4182_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,times_3573771949741848930nteger(A,B)) ).

% mult_minus_left
tff(fact_4183_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_4184_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_4185_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_4186_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_4187_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_4188_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_4189_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_4190_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_4191_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_4192_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_4193_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_4194_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_4195_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_4196_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_4197_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_4198_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_4199_minus__dvd__iff,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X2)),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Y)) ) ).

% minus_dvd_iff
tff(fact_4200_minus__dvd__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X2)),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Y)) ) ).

% minus_dvd_iff
tff(fact_4201_minus__dvd__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,X2)),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),Y)) ) ).

% minus_dvd_iff
tff(fact_4202_minus__dvd__iff,axiom,
    ! [X2: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X2)),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X2),Y)) ) ).

% minus_dvd_iff
tff(fact_4203_dvd__minus__iff,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),aa_int_int(uminus_uminus_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Y)) ) ).

% dvd_minus_iff
tff(fact_4204_dvd__minus__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),aa_real_real(uminus_uminus_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Y)) ) ).

% dvd_minus_iff
tff(fact_4205_dvd__minus__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),aa_rat_rat(uminus_uminus_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),Y)) ) ).

% dvd_minus_iff
tff(fact_4206_dvd__minus__iff,axiom,
    ! [X2: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X2),Y)) ) ).

% dvd_minus_iff
tff(fact_4207_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_4208_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_4209_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_4210_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_4211_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),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),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_4212_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_4213_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4214_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4215_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4216_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_4217_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4218_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4219_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat(fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4220_of__bool__eq_I1_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4221_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = zero_zero_complex )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4222_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = zero_zero_real )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4223_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4224_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = zero_zero_nat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4225_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = zero_zero_int )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_4226_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4227_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4228_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4229_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_4230_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = one_one_complex )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4231_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n3304061248610475627l_real(P) = one_one_real )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4232_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = one_one_rat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4233_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = one_one_nat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4234_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = one_one_int )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_4235_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex(fTrue) = one_one_complex ).

% of_bool_eq(2)
tff(fact_4236_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real(fTrue) = one_one_real ).

% of_bool_eq(2)
tff(fact_4237_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat(fTrue) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4238_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat(fTrue) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4239_of__bool__eq_I2_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fTrue) = one_one_int ).

% of_bool_eq(2)
tff(fact_4240_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_or_iff
tff(fact_4241_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(P),Q)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_or_iff
tff(fact_4242_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_4243_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_4244_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_4245_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_4246_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_4247_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_4248_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_4249_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4250_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_4251_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_4252_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_4253_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_4254_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_4255_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_4256_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_4257_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_le_0_iff_le
tff(fact_4258_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_4259_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_4260_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_4261_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_4262_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_4263_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_4264_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_4265_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_4266_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_4267_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_0_iff_less
tff(fact_4268_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_4269_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_4270_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_4271_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_4272_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_4273_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_4274_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_4275_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_pos
tff(fact_4276_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_4277_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_4278_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_4279_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_4280_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_4281_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_4282_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_4283_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_4284_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_4285_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_4286_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_4287_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_4288_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_4289_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_4290_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_4291_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_4292_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_4293_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_4294_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_4295_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_4296_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_4297_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4298_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_4299_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_4300_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_4301_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_4302_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4303_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4304_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4305_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4306_right__diff__distrib__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(V),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V),B),times_3573771949741848930nteger(numera6620942414471956472nteger(V),C)) ).

% right_diff_distrib_numeral
tff(fact_4307_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_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))) ).

% left_diff_distrib_numeral
tff(fact_4308_left__diff__distrib__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(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_4309_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_4310_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B),numera6620942414471956472nteger(V)) = minus_8373710615458151222nteger(times_3573771949741848930nteger(A,numera6620942414471956472nteger(V)),times_3573771949741848930nteger(B,numera6620942414471956472nteger(V))) ).

% left_diff_distrib_numeral
tff(fact_4311_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_4312_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_4313_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_4314_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_4315_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4316_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4317_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_4318_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_4319_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_4320_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4321_add__neg__numeral__simps_I3_J,axiom,
    ! [M: 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,M))),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,M)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4322_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),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,M)),aa_num_real(numeral_numeral_real,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4323_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),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,M)),aa_num_rat(numeral_numeral_rat,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4324_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).

% add_neg_numeral_simps(3)
tff(fact_4325_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_4326_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_4327_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_4328_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_4329_mult__minus1__right,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(Z,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_4330_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_4331_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_4332_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_4333_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_4334_mult__minus1,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_4335_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_4336_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_4337_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_4338_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_4339_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_4340_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_4341_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_4342_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_4343_divide__minus1,axiom,
    ! [X2: complex] : aa_complex_complex(divide1717551699836669952omplex(X2),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X2) ).

% divide_minus1
tff(fact_4344_divide__minus1,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X2) ).

% divide_minus1
tff(fact_4345_divide__minus1,axiom,
    ! [X2: rat] : aa_rat_rat(divide_divide_rat(X2),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X2) ).

% divide_minus1
tff(fact_4346_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_diff
tff(fact_4347_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4348_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4349_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4350_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_4351_zero__less__diff,axiom,
    ! [N: nat,M: 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),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_4352_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_4353_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_4354_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),one_one_real))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4355_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4356_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4357_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_4358_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fNot,P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex(P)) ).

% of_bool_not_iff
tff(fact_4359_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fNot,P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real(P)) ).

% of_bool_not_iff
tff(fact_4360_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat(P)) ).

% of_bool_not_iff
tff(fact_4361_of__bool__not__iff,axiom,
    ! [P: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_bool_int(zero_n2684676970156552555ol_int,P)) ).

% of_bool_not_iff
tff(fact_4362_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_4363_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_4364_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_4365_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_4366_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) = 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_4367_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_4368_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_simps(1)
tff(fact_4369_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_simps(1)
tff(fact_4370_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_4371_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_4372_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_4373_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_4374_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_4375_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_4376_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_4377_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_4378_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_4379_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_4380_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_4381_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_4382_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_4383_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_4384_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_4385_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_4386_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4387_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4388_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4389_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4390_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4391_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4392_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4393_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4394_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4395_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4396_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_4397_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4398_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_4399_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(4)
tff(fact_4400_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_4401_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_4402_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_4403_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(3)
tff(fact_4404_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_4405_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_4406_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_4407_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(2)
tff(fact_4408_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_4409_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_4410_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_4411_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_4412_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4413_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_4414_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_4415_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_4416_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W2)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W2))),Y) ).

% semiring_norm(168)
tff(fact_4417_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_4418_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_4419_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_4420_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),N)) ).

% diff_numeral_simps(2)
tff(fact_4421_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_4422_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_4423_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_4424_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),N))) ).

% diff_numeral_simps(3)
tff(fact_4425_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4426_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4427_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4428_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(times_times_num(M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4429_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4430_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4431_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4432_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4433_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4434_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4435_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4436_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : times_3573771949741848930nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4437_neg__numeral__le__iff,axiom,
    ! [M: 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,M))),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),M)) ) ).

% neg_numeral_le_iff
tff(fact_4438_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4439_neg__numeral__le__iff,axiom,
    ! [M: 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,M))),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),M)) ) ).

% neg_numeral_le_iff
tff(fact_4440_neg__numeral__le__iff,axiom,
    ! [M: 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,M))),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),M)) ) ).

% neg_numeral_le_iff
tff(fact_4441_neg__numeral__less__iff,axiom,
    ! [M: 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,M))),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),M)) ) ).

% neg_numeral_less_iff
tff(fact_4442_neg__numeral__less__iff,axiom,
    ! [M: 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,M))),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),M)) ) ).

% neg_numeral_less_iff
tff(fact_4443_neg__numeral__less__iff,axiom,
    ! [M: 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,M))),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),M)) ) ).

% neg_numeral_less_iff
tff(fact_4444_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4445_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: 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,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4446_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4447_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: 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,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4448_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: 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,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4449_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_4450_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_4451_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_4452_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_4453_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_4454_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_4455_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_4456_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_4457_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_4458_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_4459_neg__numeral__less__neg__one__iff,axiom,
    ! [M: 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,M))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4460_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4461_neg__numeral__less__neg__one__iff,axiom,
    ! [M: 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,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4462_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4463_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_4464_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_4465_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_4466_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_4467_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_4468_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_4469_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_4470_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_4471_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_4472_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4473_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4474_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4475_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4476_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4477_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4478_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4479_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4480_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4481_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4482_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4483_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_4484_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ).

% even_diff
tff(fact_4485_of__bool__half__eq__0,axiom,
    ! [B: bool] : divide6298287555418463151nteger(zero_n356916108424825756nteger(B),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4486_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_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_4487_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_4488_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4489_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4490_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4491_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4492_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),N)) ).

% diff_numeral_special(3)
tff(fact_4493_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4494_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4495_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4496_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4497_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_4498_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4499_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4500_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4501_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4502_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4503_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_4504_even__diff__nat,axiom,
    ! [M: 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,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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,M),N))) ) ) ).

% even_diff_nat
tff(fact_4505_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4506_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_4507_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_4508_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_4509_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4510_bits__1__div__exp,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_4511_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)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_4512_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_4513_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_4514_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)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_4515_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_4516_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_4517_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_4518_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_4519_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4520_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_4521_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_4522_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_4523_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_4524_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4525_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4526_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4527_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_4528_of__bool__eq__iff,axiom,
    ! [P3: bool,Q2: bool] :
      ( ( zero_n2687167440665602831ol_nat(P3) = zero_n2687167440665602831ol_nat(Q2) )
    <=> ( pp(P3)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4529_of__bool__eq__iff,axiom,
    ! [P3: bool,Q2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = aa_bool_int(zero_n2684676970156552555ol_int,Q2) )
    <=> ( pp(P3)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4530_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4531_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4532_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4533_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4534_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_4535_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% diff_right_commute
tff(fact_4536_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_4537_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% diff_right_commute
tff(fact_4538_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% diff_right_commute
tff(fact_4539_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_4540_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_4541_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_4542_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_4543_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_4544_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_4545_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_4546_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_4547_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_4548_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_4549_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_4550_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_4551_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4552_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4553_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4554_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fconj(P),Q)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)) ).

% of_bool_conj
tff(fact_4555_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fconj(P),Q)) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)) ).

% of_bool_conj
tff(fact_4556_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_conj
tff(fact_4557_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(P),Q)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_conj
tff(fact_4558_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: 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),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_mono
tff(fact_4559_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_mono
tff(fact_4560_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: 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),C))
       => 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),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_mono
tff(fact_4561_diff__left__mono,axiom,
    ! [B: real,A: real,C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_left_mono
tff(fact_4562_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: 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(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_left_mono
tff(fact_4563_diff__left__mono,axiom,
    ! [B: int,A: int,C: 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,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_left_mono
tff(fact_4564_diff__right__mono,axiom,
    ! [A: real,B: real,C: 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),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_right_mono
tff(fact_4565_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_right_mono
tff(fact_4566_diff__right__mono,axiom,
    ! [A: int,B: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_right_mono
tff(fact_4567_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4568_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( 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,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4569_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( 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,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4570_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_4571_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_4572_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_4573_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_4574_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_strict_mono
tff(fact_4575_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: 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),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_strict_mono
tff(fact_4576_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: 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),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_strict_mono
tff(fact_4577_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_4578_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( 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,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_4579_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( 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,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_4580_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: 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,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_strict_left_mono
tff(fact_4581_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: 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(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_strict_left_mono
tff(fact_4582_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: 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,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_strict_left_mono
tff(fact_4583_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: 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),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_strict_right_mono
tff(fact_4584_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: 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),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_strict_right_mono
tff(fact_4585_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_strict_right_mono
tff(fact_4586_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_4587_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_minus_iff
tff(fact_4588_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_4589_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_4590_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_4591_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_le_iff
tff(fact_4592_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_4593_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_4594_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_4595_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_imp_neg_le
tff(fact_4596_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_4597_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_4598_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_4599_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_4600_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_4601_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_4602_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_4603_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_4604_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_4605_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_4606_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_4607_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_4608_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_4609_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_4610_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_4611_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_4612_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_4613_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_4614_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_4615_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% less_minus_iff
tff(fact_4616_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_4617_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_4618_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_4619_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_less_iff
tff(fact_4620_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_4621_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_4622_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_4623_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4624_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4625_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4626_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4627_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_4628_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_4629_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_4630_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4631_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4632_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4633_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_4634_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).

% add_diff_eq
tff(fact_4635_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_4636_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_4637_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_diff_eq2
tff(fact_4638_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_4639_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_4640_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_add_eq
tff(fact_4641_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_4642_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4643_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4644_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4645_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_4646_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_4647_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4648_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_4649_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_4650_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_4651_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_4652_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_4653_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_4654_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_4655_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_4656_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% numeral_neq_neg_numeral
tff(fact_4657_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_real(numeral_numeral_real,M) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% numeral_neq_neg_numeral
tff(fact_4658_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_neq_neg_numeral
tff(fact_4659_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6620942414471956472nteger(M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% numeral_neq_neg_numeral
tff(fact_4660_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ).

% neg_numeral_neq_numeral
tff(fact_4661_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) != aa_num_real(numeral_numeral_real,N) ).

% neg_numeral_neq_numeral
tff(fact_4662_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,N) ).

% neg_numeral_neq_numeral
tff(fact_4663_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(N) ).

% neg_numeral_neq_numeral
tff(fact_4664_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_4665_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_4666_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_4667_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = times_3573771949741848930nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_4668_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_4669_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_4670_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_4671_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger(A,A) = times_3573771949741848930nteger(B,B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_4672_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_4673_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_4674_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_4675_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_4676_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_4677_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_4678_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_4679_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4680_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4681_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_4682_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_4683_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_4684_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_4685_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_4686_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_4687_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_4688_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_4689_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_4690_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_4691_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_4692_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_4693_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_4694_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_4695_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_4696_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_4697_dvd__diff,axiom,
    ! [X2: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))) ) ) ).

% dvd_diff
tff(fact_4698_dvd__diff,axiom,
    ! [X2: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X2),aa_rat_rat(minus_minus_rat(Y),Z))) ) ) ).

% dvd_diff
tff(fact_4699_dvd__diff,axiom,
    ! [X2: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z))) ) ) ).

% dvd_diff
tff(fact_4700_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_4701_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_4702_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_4703_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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),M))) ) ) ).

% diff_less_mono2
tff(fact_4704_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => 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)),K)) ) ).

% less_imp_diff_less
tff(fact_4705_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_4706_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_4707_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4708_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_4709_diff__le__self,axiom,
    ! [M: 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,M),N)),M)) ).

% diff_le_self
tff(fact_4710_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: 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,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_4711_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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),M))) ) ).

% diff_le_mono2
tff(fact_4712_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_4713_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: 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,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_4714_diff__cancel2,axiom,
    ! [M: nat,K: 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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_4715_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

% diff_add_inverse
tff(fact_4716_diff__add__inverse2,axiom,
    ! [M: 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,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_4717_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% diff_mult_distrib2
tff(fact_4718_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% diff_mult_distrib
tff(fact_4719_max__diff__distrib__left,axiom,
    ! [X2: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X2),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_4720_max__diff__distrib__left,axiom,
    ! [X2: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(X2),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(X2),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_4721_max__diff__distrib__left,axiom,
    ! [X2: 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,X2),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_4722_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_4723_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_4724_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_4725_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_4726_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_4727_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_4728_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_4729_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X2),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X2)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4730_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X2: 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,X2),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,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4731_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: 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(X2),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X2)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4732_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4733_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4734_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4735_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4736_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4737_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real(P))) ).

% zero_less_eq_of_bool
tff(fact_4738_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P))) ).

% zero_less_eq_of_bool
tff(fact_4739_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P))) ).

% zero_less_eq_of_bool
tff(fact_4740_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P))) ).

% zero_less_eq_of_bool
tff(fact_4741_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P)),one_one_real)) ).

% of_bool_less_eq_one
tff(fact_4742_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat)) ).

% of_bool_less_eq_one
tff(fact_4743_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat)) ).

% of_bool_less_eq_one
tff(fact_4744_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int)) ).

% of_bool_less_eq_one
tff(fact_4745_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = one_one_complex ) )
      & ( ~ pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = zero_zero_complex ) ) ) ).

% of_bool_def
tff(fact_4746_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n3304061248610475627l_real(P3) = one_one_real ) )
      & ( ~ pp(P3)
       => ( zero_n3304061248610475627l_real(P3) = zero_zero_real ) ) ) ).

% of_bool_def
tff(fact_4747_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n2052037380579107095ol_rat(P3) = one_one_rat ) )
      & ( ~ pp(P3)
       => ( zero_n2052037380579107095ol_rat(P3) = zero_zero_rat ) ) ) ).

% of_bool_def
tff(fact_4748_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n2687167440665602831ol_nat(P3) = one_one_nat ) )
      & ( ~ pp(P3)
       => ( zero_n2687167440665602831ol_nat(P3) = zero_zero_nat ) ) ) ).

% of_bool_def
tff(fact_4749_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = one_one_int ) )
      & ( ~ pp(P3)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = zero_zero_int ) ) ) ).

% of_bool_def
tff(fact_4750_split__of__bool,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ( ( pp(P3)
         => pp(aa_complex_bool(P,one_one_complex)) )
        & ( ~ pp(P3)
         => pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool
tff(fact_4751_split__of__bool,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P3)))
    <=> ( ( pp(P3)
         => pp(aa_real_bool(P,one_one_real)) )
        & ( ~ pp(P3)
         => pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool
tff(fact_4752_split__of__bool,axiom,
    ! [P: fun_rat_bool,P3: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P3)))
    <=> ( ( pp(P3)
         => pp(aa_rat_bool(P,one_one_rat)) )
        & ( ~ pp(P3)
         => pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool
tff(fact_4753_split__of__bool,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P3)))
    <=> ( ( pp(P3)
         => pp(aa_nat_bool(P,one_one_nat)) )
        & ( ~ pp(P3)
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool
tff(fact_4754_split__of__bool,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P3)))
    <=> ( ( pp(P3)
         => pp(aa_int_bool(P,one_one_int)) )
        & ( ~ pp(P3)
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool
tff(fact_4755_split__of__bool__asm,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_complex_bool(P,one_one_complex)) )
          | ( ~ pp(P3)
            & ~ pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool_asm
tff(fact_4756_split__of__bool__asm,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_real_bool(P,one_one_real)) )
          | ( ~ pp(P3)
            & ~ pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool_asm
tff(fact_4757_split__of__bool__asm,axiom,
    ! [P: fun_rat_bool,P3: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_rat_bool(P,one_one_rat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool_asm
tff(fact_4758_split__of__bool__asm,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_nat_bool(P,one_one_nat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool_asm
tff(fact_4759_split__of__bool__asm,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_int_bool(P,one_one_int)) )
          | ( ~ pp(P3)
            & ~ pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool_asm
tff(fact_4760_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_4761_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_4762_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_4763_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_4764_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_4765_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_4766_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_le_eq
tff(fact_4767_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% diff_le_eq
tff(fact_4768_diff__le__eq,axiom,
    ! [A: int,B: int,C: 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)),C))
    <=> 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,C),B))) ) ).

% diff_le_eq
tff(fact_4769_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% le_diff_eq
tff(fact_4770_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C)) ) ).

% le_diff_eq
tff(fact_4771_le__diff__eq,axiom,
    ! [A: int,C: 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,C),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)),C)) ) ).

% le_diff_eq
tff(fact_4772_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_4773_le__add__diff,axiom,
    ! [A: nat,B: nat,C: 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,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A))) ) ).

% le_add_diff
tff(fact_4774_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: 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,C),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,C),A)),B)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4775_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: 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,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4776_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: 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,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4777_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: 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)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4778_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: 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),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4779_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: 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,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4780_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_4781_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: 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) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4782_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_4783_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_4784_add__le__imp__le__diff,axiom,
    ! [I: nat,K: 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,I),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_4785_add__le__imp__le__diff,axiom,
    ! [I: int,K: 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,I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_4786_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: 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,I),K)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4787_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4788_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: 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,I),K)),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),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),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),K)))
           => 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),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4789_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: 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,I),K)),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),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),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),K)))
           => 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),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4790_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_less_eq
tff(fact_4791_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% diff_less_eq
tff(fact_4792_diff__less__eq,axiom,
    ! [A: int,B: int,C: 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)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_less_eq
tff(fact_4793_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% less_diff_eq
tff(fact_4794_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C)) ) ).

% less_diff_eq
tff(fact_4795_less__diff__eq,axiom,
    ! [A: int,C: 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,C),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)),C)) ) ).

% less_diff_eq
tff(fact_4796_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_4797_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_4798_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_4799_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_4800_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4801_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4802_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4803_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4804_neg__numeral__le__numeral,axiom,
    ! [M: 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,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_le_numeral
tff(fact_4805_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_le_numeral
tff(fact_4806_neg__numeral__le__numeral,axiom,
    ! [M: 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,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_le_numeral
tff(fact_4807_neg__numeral__le__numeral,axiom,
    ! [M: 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,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_le_numeral
tff(fact_4808_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_4809_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_4810_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_4811_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_4812_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% zero_neq_neg_numeral
tff(fact_4813_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4814_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4815_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4816_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4817_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4818_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4819_square__diff__square__factored,axiom,
    ! [X2: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),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,X2),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y)) ).

% square_diff_square_factored
tff(fact_4820_square__diff__square__factored,axiom,
    ! [X2: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),X2)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X2),Y)),aa_rat_rat(minus_minus_rat(X2),Y)) ).

% square_diff_square_factored
tff(fact_4821_square__diff__square__factored,axiom,
    ! [X2: 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,X2),X2)),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,X2),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),Y)) ).

% square_diff_square_factored
tff(fact_4822_mult__diff__mult,axiom,
    ! [X2: 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,X2),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,X2),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,X2),A)),B)) ).

% mult_diff_mult
tff(fact_4823_mult__diff__mult,axiom,
    ! [X2: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X2),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),A)),B)) ).

% mult_diff_mult
tff(fact_4824_mult__diff__mult,axiom,
    ! [X2: 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,X2),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,X2),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,X2),A)),B)) ).

% mult_diff_mult
tff(fact_4825_neg__numeral__less__numeral,axiom,
    ! [M: 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,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_less_numeral
tff(fact_4826_neg__numeral__less__numeral,axiom,
    ! [M: 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,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_less_numeral
tff(fact_4827_neg__numeral__less__numeral,axiom,
    ! [M: 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,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_less_numeral
tff(fact_4828_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_less_numeral
tff(fact_4829_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4830_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4831_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4832_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_4833_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_4834_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_4835_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_4836_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_4837_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_4838_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_4839_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_4840_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_4841_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_4842_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_4843_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_4844_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_4845_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_4846_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_4847_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_4848_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_4849_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_4850_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4851_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_4852_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_4853_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_4854_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_4855_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4856_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_4857_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_4858_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_4859_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_4860_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_4861_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_4862_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_4863_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_4864_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_4865_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_4866_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_4867_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_4868_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_4869_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_4870_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_4871_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_4872_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_4873_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_4874_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_4875_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_4876_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_4877_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_4878_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_4879_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_4880_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_4881_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_4882_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_4883_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_4884_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_4885_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% one_neq_neg_numeral
tff(fact_4886_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% one_neq_neg_numeral
tff(fact_4887_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% one_neq_neg_numeral
tff(fact_4888_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% one_neq_neg_numeral
tff(fact_4889_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% one_neq_neg_numeral
tff(fact_4890_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_4891_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_4892_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,N) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_4893_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_4894_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6620942414471956472nteger(N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_4895_square__eq__1__iff,axiom,
    ! [X2: complex] :
      ( ( aa_complex_complex(times_times_complex(X2),X2) = one_one_complex )
    <=> ( ( X2 = one_one_complex )
        | ( X2 = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_4896_square__eq__1__iff,axiom,
    ! [X2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2) = one_one_int )
    <=> ( ( X2 = one_one_int )
        | ( X2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_4897_square__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2) = one_one_real )
    <=> ( ( X2 = one_one_real )
        | ( X2 = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_4898_square__eq__1__iff,axiom,
    ! [X2: rat] :
      ( ( aa_rat_rat(times_times_rat(X2),X2) = one_one_rat )
    <=> ( ( X2 = one_one_rat )
        | ( X2 = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_4899_square__eq__1__iff,axiom,
    ! [X2: code_integer] :
      ( ( times_3573771949741848930nteger(X2,X2) = one_one_Code_integer )
    <=> ( ( X2 = one_one_Code_integer )
        | ( X2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_4900_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_4901_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_4902_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_4903_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_4904_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_4905_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_4906_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_4907_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_4908_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_4909_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_4910_diff__less__Suc,axiom,
    ! [M: 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,M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_4911_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_4912_diff__less,axiom,
    ! [N: nat,M: 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),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_4913_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_4914_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_4915_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: 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,C),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),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).

% diff_less_mono
tff(fact_4916_diff__add__0,axiom,
    ! [N: nat,M: 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),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_4917_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).

% less_diff_conv
tff(fact_4918_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_4919_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: 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),K)),I))
    <=> 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,I),K))) ) ).

% le_diff_conv
tff(fact_4920_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_4921_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4922_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4923_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4924_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),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,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_4925_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_4926_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_4927_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_4928_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_4929_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X4: nat,Y2: 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(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),Y2)) = D4 )
        | ( 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),Y2)) = D4 ) ) ) ).

% bezout1_nat
tff(fact_4930_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_4931_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = 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_4932_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

% mod_if
tff(fact_4933_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_4934_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_4935_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_4936_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: 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)),C)),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)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4937_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: 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)),C)),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)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4938_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> 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)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4939_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: 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)),C)),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,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4940_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: 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)),C)),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,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4941_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4942_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_4943_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_4944_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_4945_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_4946_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_4947_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_le_neg_numeral
tff(fact_4948_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_4949_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_4950_less__add__iff1,axiom,
    ! [A: real,E2: real,C: 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)),C)),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)),C)),D)) ) ).

% less_add_iff1
tff(fact_4951_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: 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)),C)),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)),C)),D)) ) ).

% less_add_iff1
tff(fact_4952_less__add__iff1,axiom,
    ! [A: int,E2: int,C: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> 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)),C)),D)) ) ).

% less_add_iff1
tff(fact_4953_less__add__iff2,axiom,
    ! [A: real,E2: real,C: 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)),C)),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,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4954_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: 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)),C)),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,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4955_less__add__iff2,axiom,
    ! [A: int,E2: int,C: 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)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_4956_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_4957_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_4958_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_4959_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_4960_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_4961_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_4962_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_4963_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_less_neg_numeral
tff(fact_4964_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_4965_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_4966_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_4967_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_4968_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_4969_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_4970_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_4971_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_4972_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_4973_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_4974_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_4975_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X2: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X2),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X2),Z)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4976_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X2),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(X2),Z)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4977_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X2: 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,X2),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,X2),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_4978_diff__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X2),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4979_diff__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X2),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(X2),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4980_diff__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),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,X2),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4981_divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X2),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4982_divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X2),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4983_divide__diff__eq__iff,axiom,
    ! [Z: real,X2: 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,X2),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4984_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_4985_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_4986_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_4987_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_4988_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_4989_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_4990_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_4991_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_4992_square__diff__one__factored,axiom,
    ! [X2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X2),X2)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X2),one_one_complex)),aa_complex_complex(minus_minus_complex(X2),one_one_complex)) ).

% square_diff_one_factored
tff(fact_4993_square__diff__one__factored,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),X2)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)) ).

% square_diff_one_factored
tff(fact_4994_square__diff__one__factored,axiom,
    ! [X2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X2),X2)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X2),one_one_rat)),aa_rat_rat(minus_minus_rat(X2),one_one_rat)) ).

% square_diff_one_factored
tff(fact_4995_square__diff__one__factored,axiom,
    ! [X2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X2),X2)),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,X2),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),one_one_int)) ).

% square_diff_one_factored
tff(fact_4996_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_4997_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_4998_neg__numeral__le__one,axiom,
    ! [M: 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,M))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_4999_neg__numeral__le__one,axiom,
    ! [M: 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,M))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_5000_neg__one__le__numeral,axiom,
    ! [M: 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,M))) ).

% neg_one_le_numeral
tff(fact_5001_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_le_numeral
tff(fact_5002_neg__one__le__numeral,axiom,
    ! [M: 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,M))) ).

% neg_one_le_numeral
tff(fact_5003_neg__one__le__numeral,axiom,
    ! [M: 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,M))) ).

% neg_one_le_numeral
tff(fact_5004_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_5005_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_5006_neg__numeral__le__neg__one,axiom,
    ! [M: 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,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_5007_neg__numeral__le__neg__one,axiom,
    ! [M: 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,M))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_5008_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_5009_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_5010_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_5011_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_5012_not__one__le__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_one_le_neg_numeral
tff(fact_5013_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_le_neg_numeral
tff(fact_5014_not__one__le__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_one_le_neg_numeral
tff(fact_5015_not__one__le__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_one_le_neg_numeral
tff(fact_5016_neg__numeral__less__one,axiom,
    ! [M: 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,M))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_5017_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_5018_neg__numeral__less__one,axiom,
    ! [M: 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,M))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_5019_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_5020_neg__one__less__numeral,axiom,
    ! [M: 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,M))) ).

% neg_one_less_numeral
tff(fact_5021_neg__one__less__numeral,axiom,
    ! [M: 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,M))) ).

% neg_one_less_numeral
tff(fact_5022_neg__one__less__numeral,axiom,
    ! [M: 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,M))) ).

% neg_one_less_numeral
tff(fact_5023_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_less_numeral
tff(fact_5024_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_5025_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_5026_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_5027_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_5028_not__one__less__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_one_less_neg_numeral
tff(fact_5029_not__one__less__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_one_less_neg_numeral
tff(fact_5030_not__one__less__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_one_less_neg_numeral
tff(fact_5031_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_less_neg_numeral
tff(fact_5032_not__neg__one__less__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5033_not__neg__one__less__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5034_not__neg__one__less__neg__numeral,axiom,
    ! [M: 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,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5035_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5036_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5037_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5038_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5039_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5040_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5041_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_5042_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5043_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5044_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5045_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5046_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5047_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5048_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_5049_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_5050_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_5051_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_5052_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_5053_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_5054_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_5055_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_5056_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_5057_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_5058_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : times_3573771949741848930nteger(B,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_5059_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_5060_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_5061_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_5062_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_5063_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_5064_inf__period_I4_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D5))
     => ! [X: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_5065_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))
     => ! [X: rat,K4: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),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(X),aa_rat_rat(times_times_rat(K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_5066_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))
     => ! [X: 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,X),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,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_5067_inf__period_I3_J,axiom,
    ! [D: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D5))
     => ! [X: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),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,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_5068_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))
     => ! [X: rat,K4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),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(X),aa_rat_rat(times_times_rat(K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_5069_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))
     => ! [X: 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,X),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,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_5070_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_5071_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_5072_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_5073_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_5074_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_5075_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_5076_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_5077_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_5078_diff__Suc__less,axiom,
    ! [N: nat,I: 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,I))),N)) ) ).

% diff_Suc_less
tff(fact_5079_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,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(P,zero_zero_nat)) )
        & ! [D6: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
           => pp(aa_nat_bool(P,D6)) ) ) ) ).

% nat_diff_split
tff(fact_5080_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,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(P,zero_zero_nat)) )
          | ? [D6: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
              & ~ pp(aa_nat_bool(P,D6)) ) ) ) ).

% nat_diff_split_asm
tff(fact_5081_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_5082_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = 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,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_5083_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = 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) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_5084_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( 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,I),U)),M)),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,I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_5085_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),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,I),U)),M)),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,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_5086_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),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,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_5087_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),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,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_5088_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5089_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),zero_z3403309356797280102nteger))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_5090_exp__div__exp__eq,axiom,
    ! [M: 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))),M)),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,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))),M)),zero_zero_nat))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_5091_exp__div__exp__eq,axiom,
    ! [M: 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))),M)),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))),M)),zero_zero_int))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),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,M),N))) ).

% exp_div_exp_eq
tff(fact_5092_frac__le__eq,axiom,
    ! [Y: real,Z: real,X2: 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,X2),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,X2),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_5093_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X2: 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(X2),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(X2),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_5094_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X2: 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(X2),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(X2),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_5095_frac__less__eq,axiom,
    ! [Y: real,Z: real,X2: 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,X2),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,X2),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_5096_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( 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),C))),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),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5097_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( 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),C))),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),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5098_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( 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),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5099_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( 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),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5100_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5101_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5102_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C))))
      <=> 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),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5103_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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),C))))
      <=> 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),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5104_minus__divide__less__eq,axiom,
    ! [B: real,C: 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),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_5105_minus__divide__less__eq,axiom,
    ! [B: rat,C: 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),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5106_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: 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),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_5107_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: 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),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5108_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5109_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) ) )
        & ( ( C = 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_5110_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) ) )
        & ( ( C = 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_5111_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5112_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) = B ) )
        & ( ( C = 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_5113_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) = B ) )
        & ( ( C = 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_5114_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X2),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X2)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5115_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X2: 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,X2),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,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5116_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X2: 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(X2),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,X2)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5117_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_5118_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_5119_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_5120_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_5121_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% power_diff
tff(fact_5122_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_diff
tff(fact_5123_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_5124_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)) ) ) ) ).

% power_diff
tff(fact_5125_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_5126_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: 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,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_5127_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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,M),N)),N)) ) ) ) ).

% div_if
tff(fact_5128_div__geq,axiom,
    ! [N: nat,M: 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,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_5129_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),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,M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_5130_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( 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,I),U)),M)),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,I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_5131_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),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,I),U)),M)),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,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_5132_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),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,M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_5133_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: 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),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),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,M),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),M)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_5134_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: 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),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_5135_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( 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(B2),N)),aa_int_int(divide_divide_int(A2),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_5136_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_5137_bot__empty__eq2,axiom,
    ! [X: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bot_bo6560894463593755925t_bool,X),Xa))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_5138_bot__empty__eq2,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(bot_bo8694918001306130243t_bool,X),Xa))
    <=> member373505688050248522BT_nat(produc738532404422230701BT_nat(X,Xa),bot_bo1642239108664514429BT_nat) ) ).

% bot_empty_eq2
tff(fact_5139_bot__empty__eq2,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(bot_bo6207810721336151133t_bool,X),Xa))
    <=> member5262025264175285858nt_int(product_Pair_int_int(X,Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_5140_bot__empty__eq2,axiom,
    ! [X: code_integer,Xa: bool] :
      ( pp(aa_bool_bool(aa_Cod8494480694608518558l_bool(bot_bo8679169850809196913l_bool,X),Xa))
    <=> member7941544277139469958r_bool(produc5135875973608410785r_bool(X,Xa),bot_bo3208612889749783737r_bool) ) ).

% bot_empty_eq2
tff(fact_5141_bot__empty__eq2,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(bot_bo7217057988103639803r_bool,X),Xa))
    <=> member157494554546826820nteger(produc1086072967326762835nteger(X,Xa),bot_bo4276436098303576167nteger) ) ).

% bot_empty_eq2
tff(fact_5142_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_5143_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_5144_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: 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),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_5145_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: 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),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5146_minus__divide__le__eq,axiom,
    ! [B: real,C: 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),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_5147_minus__divide__le__eq,axiom,
    ! [B: rat,C: 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),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5148_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C))))
      <=> 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),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5149_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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),C))))
      <=> 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),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5150_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5151_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5152_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( 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),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5153_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( 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),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5154_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( 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),C))),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),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5155_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( 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),C))),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),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5156_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: 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),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_5157_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: 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),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5158_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_5159_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),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),C))
         => 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))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5160_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_5161_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_5162_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: 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),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_5163_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: 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),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_5164_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_5165_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),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,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),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),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_5166_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),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,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),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),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_5167_power__eq__if,axiom,
    ! [M: nat,P3: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_5168_power__eq__if,axiom,
    ! [M: nat,P3: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_nat_real(power_power_real(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_5169_power__eq__if,axiom,
    ! [M: nat,P3: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = aa_rat_rat(times_times_rat(P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_5170_power__eq__if,axiom,
    ! [M: nat,P3: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_5171_power__eq__if,axiom,
    ! [M: nat,P3: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_5172_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_5173_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(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(power_power_real(A),N) ) ) ).

% power_minus_mult
tff(fact_5174_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_5175_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_5176_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_5177_diff__le__diff__pow,axiom,
    ! [K: nat,M: 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))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))) ) ).

% diff_le_diff_pow
tff(fact_5178_le__div__geq,axiom,
    ! [N: nat,M: 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),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_5179_bits__induct,axiom,
    ! [P: fun_Co8822979462182099027r_bool,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( divide6298287555418463151nteger(A3,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
         => pp(aa_Code_integer_bool(P,A3)) )
     => ( ! [A3: code_integer,B3: bool] :
            ( pp(aa_Code_integer_bool(P,A3))
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B3),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A3)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
             => pp(aa_Code_integer_bool(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B3),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A3)))) ) )
       => pp(aa_Code_integer_bool(P,A)) ) ) ).

% bits_induct
tff(fact_5180_bits__induct,axiom,
    ! [P: 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(P,A3)) )
     => ( ! [A3: nat,B3: bool] :
            ( pp(aa_nat_bool(P,A3))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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(P,A)) ) ) ).

% bits_induct
tff(fact_5181_bits__induct,axiom,
    ! [P: 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(P,A3)) )
     => ( ! [A3: int,B3: bool] :
            ( pp(aa_int_bool(P,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(P,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(P,A)) ) ) ).

% bits_induct
tff(fact_5182_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_5183_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5184_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: 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),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => 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))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),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_5185_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: 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),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => 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))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),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_5186_square__le__1,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_5187_square__le__1,axiom,
    ! [X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X2),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)) ) ) ).

% square_le_1
tff(fact_5188_square__le__1,axiom,
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_5189_square__le__1,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_5190_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5191_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5192_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_5193_exp__mod__exp,axiom,
    ! [M: 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))),M)),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,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_5194_exp__mod__exp,axiom,
    ! [M: 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))),M)),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,M),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_5195_power2__diff,axiom,
    ! [X2: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X2),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(X2),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))),X2)),Y)) ).

% power2_diff
tff(fact_5196_power2__diff,axiom,
    ! [X2: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X2),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(X2),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))),X2)),Y)) ).

% power2_diff
tff(fact_5197_power2__diff,axiom,
    ! [X2: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),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(X2),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))),X2)),Y)) ).

% power2_diff
tff(fact_5198_power2__diff,axiom,
    ! [X2: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(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))),X2)),Y)) ).

% power2_diff
tff(fact_5199_power2__diff,axiom,
    ! [X2: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(X2,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_3573771949741848930nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),X2),Y)) ).

% power2_diff
tff(fact_5200_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(times_3573771949741848930nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = times_3573771949741848930nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5201_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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))),M))),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),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5202_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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))),M))),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),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5203_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_5204_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: 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))),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5205_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),K) = K )
    <=> ( 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))),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_5206_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B))))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5207_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_5208_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_5209_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_5210_even__mask__div__iff_H,axiom,
    ! [M: 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))),M)),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,M),N)) ) ).

% even_mask_div_iff'
tff(fact_5211_even__mask__div__iff_H,axiom,
    ! [M: 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))),M)),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,M),N)) ) ).

% even_mask_div_iff'
tff(fact_5212_option_Osize__gen_I1_J,axiom,
    ! [X2: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X2,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5213_option_Osize__gen_I1_J,axiom,
    ! [X2: fun_num_nat] : size_option_num(X2,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5214_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_5215_bot__empty__eq,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(bot_bo1239524807443452474T_bool,X))
    <=> pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),bot_bo8194388402131092736T_VEBT)) ) ).

% bot_empty_eq
tff(fact_5216_bot__empty__eq,axiom,
    ! [X: set_nat] :
      ( pp(aa_set_nat_bool(bot_bo8343409700046576086t_bool,X))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X),bot_bot_set_set_nat)) ) ).

% bot_empty_eq
tff(fact_5217_bot__empty__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(bot_bo4181092155253064880l_bool,X))
    <=> pp(aa_set_real_bool(member_real(X),bot_bot_set_real)) ) ).

% bot_empty_eq
tff(fact_5218_bot__empty__eq,axiom,
    ! [X: nat] :
      ( pp(aa_nat_bool(bot_bot_fun_nat_bool,X))
    <=> pp(aa_set_nat_bool(member_nat(X),bot_bot_set_nat)) ) ).

% bot_empty_eq
tff(fact_5219_bot__empty__eq,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(bot_bot_fun_int_bool,X))
    <=> pp(aa_set_int_bool(member_int(X),bot_bot_set_int)) ) ).

% bot_empty_eq
tff(fact_5220_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_5221_even__mask__div__iff,axiom,
    ! [M: 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))),M)),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,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_5222_even__mask__div__iff,axiom,
    ! [M: 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))),M)),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,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_5223_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = produc1086072967326762835nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q2),R2) ) ) ) ).

% divmod_step_eq
tff(fact_5224_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_5225_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_5226_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(T)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_5227_artanh__def,axiom,
    ! [X2: real] : aa_real_real(artanh_real,X2) = 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),X2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5228_compl__le__compl__iff,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X2)),uminus1532241313380277803et_int(Y)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Y),X2)) ) ).

% compl_le_compl_iff
tff(fact_5229_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_5230_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_5231_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_5232_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_5233_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_5234_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_5235_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_5236_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_5237_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_5238_finite__Diff2,axiom,
    ! [B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2)))
      <=> pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% finite_Diff2
tff(fact_5239_finite__Diff2,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2)))
      <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% finite_Diff2
tff(fact_5240_finite__Diff2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2)))
      <=> pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% finite_Diff2
tff(fact_5241_finite__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2))) ) ).

% finite_Diff
tff(fact_5242_finite__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2))) ) ).

% finite_Diff
tff(fact_5243_finite__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2))) ) ).

% finite_Diff
tff(fact_5244_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B2)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2)) ) ).

% Compl_subset_Compl_iff
tff(fact_5245_Compl__anti__mono,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(B2)),uminus1532241313380277803et_int(A2))) ) ).

% Compl_anti_mono
tff(fact_5246_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_5247_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat(A2,B2) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_5248_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).

% Diff_eq_empty_iff
tff(fact_5249_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or4548717258645045905et_nat(L,U)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),I))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5250_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(L),I))
        & pp(aa_set_int_bool(ord_less_eq_set_int(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5251_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_or633870826150836451st_rat(L,U)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5252_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I),set_or7049704709247886629st_num(L,U)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5253_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),set_or1269000886237332187st_nat(L,U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5254_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5255_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or1222579329274155063t_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_5256_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_5257_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_5258_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_5259_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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_5260_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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_5261_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_5262_ln__inj__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( ( aa_real_real(ln_ln_real,X2) = aa_real_real(ln_ln_real,Y) )
        <=> ( X2 = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_5263_ln__less__cancel__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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,X2)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5264_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or1269000886237332187st_nat(L,U))) ).

% finite_atLeastAtMost
tff(fact_5265_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_5266_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_5267_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_5268_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: 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_5269_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: 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_5270_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_5271_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_5272_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_5273_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_5274_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_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_5275_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_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_5276_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_5277_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
          & pp(aa_set_int_bool(ord_less_eq_set_int(B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5278_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,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,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5279_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,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,C),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5280_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,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,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5281_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,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,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5282_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,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,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_5283_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_5284_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_5285_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_5286_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_5287_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_5288_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_5289_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_5290_ln__le__cancel__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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,X2)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5291_ln__less__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_5292_ln__gt__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ) ).

% ln_gt_zero_iff
tff(fact_5293_ln__eq__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( aa_real_real(ln_ln_real,X2) = zero_zero_real )
      <=> ( X2 = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_5294_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5295_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_5296_ln__le__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_5297_ln__ge__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ) ).

% ln_ge_zero_iff
tff(fact_5298_ln__div,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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,X2),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X2)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_5299_Diff__infinite__finite,axiom,
    ! [T3: set_int,S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
       => ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,T3))) ) ) ).

% Diff_infinite_finite
tff(fact_5300_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T3))) ) ) ).

% Diff_infinite_finite
tff(fact_5301_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,T3))) ) ) ).

% Diff_infinite_finite
tff(fact_5302_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),C4))
       => ( minus_minus_set_nat(B2,minus_minus_set_nat(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_5303_double__diff,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
       => ( minus_minus_set_int(B2,minus_minus_set_int(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_5304_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,B2)),A2)) ).

% Diff_subset
tff(fact_5305_Diff__subset,axiom,
    ! [A2: set_int,B2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,B2)),A2)) ).

% Diff_subset
tff(fact_5306_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D5: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,D5),B2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,B2)),minus_minus_set_nat(C4,D5))) ) ) ).

% Diff_mono
tff(fact_5307_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D5: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D5),B2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,B2)),minus_minus_set_int(C4,D5))) ) ) ).

% Diff_mono
tff(fact_5308_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_5309_ln__one__minus__pos__upper__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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),X2))),aa_real_real(uminus_uminus_real,X2))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_5310_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_5311_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_5312_psubset__imp__ex__mem,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),B2))
     => ? [B3: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_5313_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ? [B3: int] : pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_5314_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2))
     => ? [B3: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(B3),minus_2163939370556025621et_nat(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_5315_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
     => ? [B3: real] : pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_5316_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2))
     => ? [B3: nat] : pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_5317_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_5318_ln__diff__le,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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,X2)),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,X2),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_5319_ln__eq__minus__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( aa_real_real(ln_ln_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real) )
       => ( X2 = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_5320_ln__add__one__self__le__self2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => 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),X2))),X2)) ) ).

% ln_add_one_self_le_self2
tff(fact_5321_ln__le__minus__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_5322_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_5323_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_5324_ln__less__self,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X2)),X2)) ) ).

% ln_less_self
tff(fact_5325_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N))
          & pp(aa_nat_bool(P,M4)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less
tff(fact_5326_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N))
         => pp(aa_nat_bool(P,M4)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less
tff(fact_5327_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2)))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_5328_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(A2)))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5329_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2)))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_5330_real__add__less__0__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X2))) ) ).

% real_add_less_0_iff
tff(fact_5331_real__0__less__add__iff,axiom,
    ! [X2: 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,X2),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X2)),Y)) ) ).

% real_0_less_add_iff
tff(fact_5332_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),set_or1269000886237332187st_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_5333_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_5334_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_5335_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_5336_ln__bound,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X2)),X2)) ) ).

% ln_bound
tff(fact_5337_ln__gt__zero__imp__gt__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5338_ln__less__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X2)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_5339_ln__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X2))) ) ).

% ln_gt_zero
tff(fact_5340_ln__mult,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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,X2),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X2)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_5341_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
          | ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
            & pp(aa_set_int_bool(ord_less_eq_set_int(B),D))
            & ( pp(aa_set_int_bool(ord_less_set_int(C),A))
              | pp(aa_set_int_bool(ord_less_set_int(B),D)) ) ) )
        & pp(aa_set_int_bool(ord_less_eq_set_int(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5342_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,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,C),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,C),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,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5343_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,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,C),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,C),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,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5344_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,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,C),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,C),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,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5345_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,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,C),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,C),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,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5346_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,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,C),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5347_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K3: 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,K3),D)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P1,X4)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P1,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_5348_plusinfinity,axiom,
    ! [D: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K3: 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,K3),D)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P2,X4)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P2,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_5349_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
           => ( pp(aa_int_bool(P,I3))
             => pp(aa_int_bool(P,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),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_5350_ln__ge__zero__imp__ge__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_5351_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,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),K))
         => ! [X: int] :
              ( pp(aa_int_bool(P,X))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_5352_mod__pos__geq,axiom,
    ! [L: int,K: 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),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_5353_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_5354_div__pos__geq,axiom,
    ! [L: int,K: 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),K))
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_5355_compl__le__swap2,axiom,
    ! [Y: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),X2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X2)),Y)) ) ).

% compl_le_swap2
tff(fact_5356_compl__le__swap1,axiom,
    ! [Y: set_int,X2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(X2)))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X2),uminus1532241313380277803et_int(Y))) ) ).

% compl_le_swap1
tff(fact_5357_compl__mono,axiom,
    ! [X2: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X2))) ) ).

% compl_mono
tff(fact_5358_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ~ ! [N2: nat] : X2 != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_5359_diff__shunt__var,axiom,
    ! [X2: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X2,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X2),Y)) ) ).

% diff_shunt_var
tff(fact_5360_diff__shunt__var,axiom,
    ! [X2: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X2,Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X2),Y)) ) ).

% diff_shunt_var
tff(fact_5361_diff__shunt__var,axiom,
    ! [X2: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X2,Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y)) ) ).

% diff_shunt_var
tff(fact_5362_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_5363_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),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(K),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(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5364_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),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(K),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(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5365_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),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(K),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(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5366_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_rat(ring_1_of_int_rat,K) = zero_zero_rat ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),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(K),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(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5367_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_5368_Bolzano,axiom,
    ! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [A3: real,B3: real,C3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),C3))
             => ( 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),C3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),C3)) ) ) ) )
       => ( ! [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(P,A3),B3)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_5369_set__decode__0,axiom,
    ! [X2: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),nat_set_decode(X2)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2)) ) ).

% set_decode_0
tff(fact_5370_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_el(L),Qr) ).

% divmod_step_def
tff(fact_5371_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_em(L),Qr) ).

% divmod_step_def
tff(fact_5372_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_en(L)),Qr) ).

% divmod_step_def
tff(fact_5373_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( 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,A2)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% even_set_encode_iff
tff(fact_5374_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B2),B2) = minus_minus_set_nat(A2,B2) ).

% Diff_idemp
tff(fact_5375_Diff__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
        & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ) ).

% Diff_iff
tff(fact_5376_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
    <=> ( pp(aa_set_int_bool(member_int(C),A2))
        & ~ pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% Diff_iff
tff(fact_5377_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% Diff_iff
tff(fact_5378_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
    <=> ( pp(aa_set_real_bool(member_real(C),A2))
        & ~ pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% Diff_iff
tff(fact_5379_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
    <=> ( pp(aa_set_nat_bool(member_nat(C),A2))
        & ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% Diff_iff
tff(fact_5380_DiffI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2))
       => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2))) ) ) ).

% DiffI
tff(fact_5381_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),A2))
     => ( ~ pp(aa_set_int_bool(member_int(C),B2))
       => pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2))) ) ) ).

% DiffI
tff(fact_5382_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))) ) ) ).

% DiffI
tff(fact_5383_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),A2))
     => ( ~ pp(aa_set_real_bool(member_real(C),B2))
       => pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2))) ) ) ).

% DiffI
tff(fact_5384_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(C),B2))
       => pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2))) ) ) ).

% DiffI
tff(fact_5385_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_5386_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_5387_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
    <=> ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% Compl_iff
tff(fact_5388_Compl__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),uminus8041839845116263051T_VEBT(A2)))
    <=> ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2)) ) ).

% Compl_iff
tff(fact_5389_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2)))
    <=> ~ pp(aa_set_int_bool(member_int(C),A2)) ) ).

% Compl_iff
tff(fact_5390_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2)))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% Compl_iff
tff(fact_5391_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2)))
    <=> ~ pp(aa_set_real_bool(member_real(C),A2)) ) ).

% Compl_iff
tff(fact_5392_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(C),A2))
     => pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2))) ) ).

% ComplI
tff(fact_5393_ComplI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),uminus8041839845116263051T_VEBT(A2))) ) ).

% ComplI
tff(fact_5394_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(C),A2))
     => pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2))) ) ).

% ComplI
tff(fact_5395_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2))
     => pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2))) ) ).

% ComplI
tff(fact_5396_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(C),A2))
     => pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2))) ) ).

% ComplI
tff(fact_5397_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or1266510415728281911st_int(L,U))) ).

% finite_atLeastAtMost_int
tff(fact_5398_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_real(ring_1_of_int_real,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n3304061248610475627l_real(P) ).

% of_int_of_bool
tff(fact_5399_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_rat(ring_1_of_int_rat,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n2052037380579107095ol_rat(P) ).

% of_int_of_bool
tff(fact_5400_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_int(ring_1_of_int_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% of_int_of_bool
tff(fact_5401_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_5402_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5403_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5404_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5405_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_5406_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_5407_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_5408_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_5409_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_5410_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_5411_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_5412_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_5413_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_5414_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_5415_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_5416_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_5417_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_5418_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_5419_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = numera6620942414471956472nteger(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5420_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_5421_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_5422_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_5423_of__int__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_num_int(numeral_numeral_int,K)) = numera6620942414471956472nteger(K) ).

% of_int_numeral
tff(fact_5424_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_5425_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_5426_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_5427_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_5428_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_5429_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_5430_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_5431_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_5432_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_5433_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_5434_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_5435_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_5436_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_5437_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_5438_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_5439_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_5440_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_5441_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_5442_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_5443_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_5444_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_5445_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_5446_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_5447_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_5448_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W2: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X2) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) )
    <=> ( X2 = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5449_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X2) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2) )
    <=> ( X2 = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5450_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X2) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) )
    <=> ( X2 = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5451_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X2) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) )
    <=> ( X2 = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5452_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: 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,X2) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5453_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: int] :
      ( ( aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2) = aa_int_real(ring_1_of_int_real,X2) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5454_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: 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,X2) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5455_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) = aa_int_complex(ring_17405671764205052669omplex,X2) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5456_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_5457_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(power_power_real(aa_int_real(ring_1_of_int_real,Z)),N) ).

% of_int_power
tff(fact_5458_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_5459_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_5460_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_5461_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_5462_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_5463_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_5464_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_5465_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_5466_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_5467_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_5468_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_5469_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_5470_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_5471_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_5472_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_5473_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_5474_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_5475_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_5476_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),numera6620942414471956472nteger(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_5477_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_5478_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_5479_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_5480_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),aa_int_Code_integer(ring_18347121197199848620nteger,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_5481_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_5482_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_5483_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_5484_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_5485_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_5486_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),numera6620942414471956472nteger(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_5487_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_5488_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_5489_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_5490_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),aa_int_Code_integer(ring_18347121197199848620nteger,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_5491_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_5492_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_5493_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_5494_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_5495_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_5496_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_5497_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_5498_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_5499_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_5500_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_5501_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_5502_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_5503_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X2)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5504_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: 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,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X2)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5505_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: 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,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X2)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5506_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),aa_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,X2),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5507_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5508_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5509_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5510_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X2)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5511_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5512_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X2)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5513_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5514_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X2)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5515_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X2)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5516_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5517_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X2)),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5518_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5519_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X2)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5520_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: 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,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X2)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5521_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X2: 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,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X2)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5522_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X2)),aa_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,X2),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5523_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5524_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5525_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X2)),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,X2)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5526_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,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,X2)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5527_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5528_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5529_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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(power_power_real(aa_num_real(numeral_numeral_real,X2)),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,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5530_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),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,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5531_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2)),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,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5532_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2)),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,X2)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5533_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5534_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5535_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X2)),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,X2)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5536_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,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,X2)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5537_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2)),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,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5538_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2)),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,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5539_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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(power_power_real(aa_num_real(numeral_numeral_real,X2)),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,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5540_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X2)),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,X2)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5541_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X2))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X2))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5542_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: 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,X2))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X2))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5543_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X2))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X2))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5544_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: 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,X2))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X2))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5545_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X2))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X2))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5546_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X2))),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,X2))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5547_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X2))),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,X2))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5548_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X2))),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,X2))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5549_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X2))),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,X2))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5550_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X2))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X2))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5551_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X2))),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,X2))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5552_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X2))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,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,X2))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5553_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: 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,X2))),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,X2))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5554_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: 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,X2))),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,X2))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5555_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X2))),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,X2))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5556_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X2))),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,X2))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5557_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2))),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,X2))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5558_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2))),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,X2))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5559_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: 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,X2))),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,X2))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5560_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X2))),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,X2))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5561_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: 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,X2))),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,X2))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5562_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X2))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,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,X2))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5563_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2))),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,X2))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5564_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X2))),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,X2))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5565_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2))),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,X2))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5566_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X2))),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,X2))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5567_Compl__eq,axiom,
    ! [A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A2) = collect_VEBT_VEBT(aTP_Lamm_eo(A2)) ).

% Compl_eq
tff(fact_5568_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_ep(A2)) ).

% Compl_eq
tff(fact_5569_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_eq(A2)) ).

% Compl_eq
tff(fact_5570_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_er(A2)) ).

% Compl_eq
tff(fact_5571_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aTP_Lamm_es(A2)) ).

% Compl_eq
tff(fact_5572_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_et(A2)) ).

% Compl_eq
tff(fact_5573_set__diff__eq,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B2) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_eu(A2),B2)) ).

% set_diff_eq
tff(fact_5574_set__diff__eq,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_ev(A2),B2)) ).

% set_diff_eq
tff(fact_5575_set__diff__eq,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_ew(A2),B2)) ).

% set_diff_eq
tff(fact_5576_set__diff__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_ex(A2),B2)) ).

% set_diff_eq
tff(fact_5577_set__diff__eq,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_ey(A2),B2)) ).

% set_diff_eq
tff(fact_5578_set__diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_ez(A2),B2)) ).

% set_diff_eq
tff(fact_5579_minus__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B2) = collect_VEBT_VEBT(minus_1873103217924207103T_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_5580_minus__set__def,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(minus_6582619742941512235l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_5581_minus__set__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(minus_8064998606961450743t_bool(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_5582_minus__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(minus_8322047804725131473t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_5583_minus__set__def,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(minus_7727386867957376939t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_5584_minus__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(minus_1628305797571225351t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% minus_set_def
tff(fact_5585_Collect__neg__eq,axiom,
    ! [P: fun_real_bool] : collect_real(aTP_Lamm_fa(P)) = uminus612125837232591019t_real(collect_real(P)) ).

% Collect_neg_eq
tff(fact_5586_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_bool] : collect_list_nat(aTP_Lamm_fb(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_5587_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_bool] : collect_set_nat(aTP_Lamm_fc(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_5588_Collect__neg__eq,axiom,
    ! [P: fun_nat_bool] : collect_nat(aTP_Lamm_fd(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_5589_Collect__neg__eq,axiom,
    ! [P: fun_int_bool] : collect_int(aTP_Lamm_fe(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_5590_uminus__set__def,axiom,
    ! [A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A2) = collect_VEBT_VEBT(uminus8258794197550912879T_bool(aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_5591_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus4638318004487240379l_bool(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_5592_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus5119773305880778631t_bool(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_5593_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus6226608070181861729t_bool(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_5594_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus1995910650241132439t_bool(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_5595_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus8094991720627284027t_bool(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_5596_DiffD2,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
     => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ).

% DiffD2
tff(fact_5597_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
     => ~ pp(aa_set_int_bool(member_int(C),B2)) ) ).

% DiffD2
tff(fact_5598_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ).

% DiffD2
tff(fact_5599_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
     => ~ pp(aa_set_real_bool(member_real(C),B2)) ) ).

% DiffD2
tff(fact_5600_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
     => ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ).

% DiffD2
tff(fact_5601_DiffD1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2)) ) ).

% DiffD1
tff(fact_5602_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
     => pp(aa_set_int_bool(member_int(C),A2)) ) ).

% DiffD1
tff(fact_5603_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
     => pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% DiffD1
tff(fact_5604_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
     => pp(aa_set_real_bool(member_real(C),A2)) ) ).

% DiffD1
tff(fact_5605_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
     => pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% DiffD1
tff(fact_5606_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),uminus5710092332889474511et_nat(A2)))
     => ~ pp(aa_set_nat_bool(member_nat(C),A2)) ) ).

% ComplD
tff(fact_5607_ComplD,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),uminus8041839845116263051T_VEBT(A2)))
     => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2)) ) ).

% ComplD
tff(fact_5608_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),uminus1532241313380277803et_int(A2)))
     => ~ pp(aa_set_int_bool(member_int(C),A2)) ) ).

% ComplD
tff(fact_5609_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),uminus613421341184616069et_nat(A2)))
     => ~ pp(aa_set_set_nat_bool(member_set_nat(C),A2)) ) ).

% ComplD
tff(fact_5610_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),uminus612125837232591019t_real(A2)))
     => ~ pp(aa_set_real_bool(member_real(C),A2)) ) ).

% ComplD
tff(fact_5611_DiffE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),minus_5127226145743854075T_VEBT(A2,B2)))
     => ~ ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),A2))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,C),B2)) ) ) ).

% DiffE
tff(fact_5612_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
     => ~ ( pp(aa_set_int_bool(member_int(C),A2))
         => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% DiffE
tff(fact_5613_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)))
     => ~ ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(C),B2)) ) ) ).

% DiffE
tff(fact_5614_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
     => ~ ( pp(aa_set_real_bool(member_real(C),A2))
         => pp(aa_set_real_bool(member_real(C),B2)) ) ) ).

% DiffE
tff(fact_5615_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
     => ~ ( pp(aa_set_nat_bool(member_nat(C),A2))
         => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% DiffE
tff(fact_5616_mult__of__int__commute,axiom,
    ! [X2: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X2)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X2)) ).

% mult_of_int_commute
tff(fact_5617_mult__of__int__commute,axiom,
    ! [X2: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X2)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X2)) ).

% mult_of_int_commute
tff(fact_5618_mult__of__int__commute,axiom,
    ! [X2: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X2)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X2)) ).

% mult_of_int_commute
tff(fact_5619_of__int__max,axiom,
    ! [X2: int,Y: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_int_real(ring_1_of_int_real,X2)),aa_int_real(ring_1_of_int_real,Y)) ).

% of_int_max
tff(fact_5620_of__int__max,axiom,
    ! [X2: int,Y: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y)) = aa_rat_rat(ord_max_rat(aa_int_rat(ring_1_of_int_rat,X2)),aa_int_rat(ring_1_of_int_rat,Y)) ).

% of_int_max
tff(fact_5621_of__int__max,axiom,
    ! [X2: int,Y: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,X2)),aa_int_int(ring_1_of_int_int,Y)) ).

% of_int_max
tff(fact_5622_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_nat_fun_nat_bool,G: fun_nat_fun_nat_bool,P3: product_prod_nat_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) = Q2 )
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,X4),Y2))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(G,X4),Y2)) ) )
     => ( ( P3 = Q2 )
       => ( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),P3))
        <=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(G),Q2)) ) ) ) ).

% split_cong
tff(fact_5623_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat,P3: product_prod_nat_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) = Q2 )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(F,X4),Y2) = aa_nat_nat(aa_nat_fun_nat_nat(G,X4),Y2) ) )
     => ( ( P3 = Q2 )
       => ( aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),P3) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(G),Q2) ) ) ) ).

% split_cong
tff(fact_5624_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P3: product_prod_nat_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) = Q2 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y2) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X4),Y2) ) )
     => ( ( P3 = Q2 )
       => ( aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),P3) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(G),Q2) ) ) ) ).

% split_cong
tff(fact_5625_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: fun_na4786964484885125043t_bool,G: fun_na4786964484885125043t_bool,P3: product_prod_nat_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2) = Q2 )
         => ( aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X4),Y2) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(G,X4),Y2) ) )
     => ( ( P3 = Q2 )
       => ( aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),P3) = aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(G),Q2) ) ) ) ).

% split_cong
tff(fact_5626_split__cong,axiom,
    ! [Q2: product_prod_int_int,F: fun_int_fun_int_bool,G: fun_int_fun_int_bool,P3: product_prod_int_int] :
      ( ! [X4: int,Y2: int] :
          ( ( product_Pair_int_int(X4,Y2) = Q2 )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(F,X4),Y2))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(G,X4),Y2)) ) )
     => ( ( P3 = Q2 )
       => ( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),P3))
        <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(G),Q2)) ) ) ) ).

% split_cong
tff(fact_5627_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,nat_set_decode(N))) ).

% finite_set_decode
tff(fact_5628_set__encode__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_5629_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
    <=> 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,M)),one_one_real))) ) ).

% int_le_real_less
tff(fact_5630_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M))
    <=> 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,M))) ) ).

% int_less_real_le
tff(fact_5631_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_5632_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_5633_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_5634_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_5635_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_5636_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_5637_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_5638_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_5639_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_5640_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_5641_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_5642_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K3: int] :
            ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))) )
       => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
        <=> ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
              & pp(aa_int_bool(P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5643_aset_I7_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D5))) ) ) ) ).

% aset(7)
tff(fact_5644_aset_I5_J,axiom,
    ! [D5: int,T: int,A2: 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),A2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D5)),T)) ) ) ) ) ).

% aset(5)
tff(fact_5645_aset_I4_J,axiom,
    ! [D5: int,T: int,A2: 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),A2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa3) ) ) )
           => ( ( X != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D5) != T ) ) ) ) ) ).

% aset(4)
tff(fact_5646_aset_I3_J,axiom,
    ! [D5: int,T: int,A2: 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)),A2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa3) ) ) )
           => ( ( X = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D5) = T ) ) ) ) ) ).

% aset(3)
tff(fact_5647_bset_I7_J,axiom,
    ! [D5: int,T: int,B2: 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),B2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D5))) ) ) ) ) ).

% bset(7)
tff(fact_5648_bset_I5_J,axiom,
    ! [D5: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D5)),T)) ) ) ) ).

% bset(5)
tff(fact_5649_bset_I4_J,axiom,
    ! [D5: int,T: int,B2: 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),B2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa3) ) ) )
           => ( ( X != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D5) != T ) ) ) ) ) ).

% bset(4)
tff(fact_5650_bset_I3_J,axiom,
    ! [D5: int,T: int,B2: 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)),B2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa3) ) ) )
           => ( ( X = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D5) = T ) ) ) ) ) ).

% bset(3)
tff(fact_5651_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_5652_aset_I8_J,axiom,
    ! [D5: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X))
           => 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,X),D5))) ) ) ) ).

% aset(8)
tff(fact_5653_aset_I6_J,axiom,
    ! [D5: int,T: int,A2: 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)),A2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),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,X),D5)),T)) ) ) ) ) ).

% aset(6)
tff(fact_5654_bset_I8_J,axiom,
    ! [D5: int,T: int,B2: 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)),B2))
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X))
             => 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,X),D5))) ) ) ) ) ).

% bset(8)
tff(fact_5655_bset_I6_J,axiom,
    ! [D5: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),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,X),D5)),T)) ) ) ) ).

% bset(6)
tff(fact_5656_cpmi,axiom,
    ! [D5: int,P: fun_int_bool,P2: fun_int_bool,B2: 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(P,X4))
            <=> pp(aa_int_bool(P2,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5)))
                 => ! [Xb2: int] :
                      ( pp(aa_set_int_bool(member_int(Xb2),B2))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5))) ) )
         => ( ! [X4: int,K3: 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,K3),D5)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),B2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5657_cppi,axiom,
    ! [D5: int,P: fun_int_bool,P2: fun_int_bool,A2: 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(P,X4))
            <=> pp(aa_int_bool(P2,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D5)))
                 => ! [Xb2: int] :
                      ( pp(aa_set_int_bool(member_int(Xb2),A2))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5))) ) )
         => ( ! [X4: int,K3: 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,K3),D5)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),A2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5658_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_en(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5659_floor__exists,axiom,
    ! [X2: real] :
    ? [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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_5660_floor__exists,axiom,
    ! [X2: 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)),X2))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),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_5661_floor__exists1,axiom,
    ! [X2: real] :
    ? [X4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y3: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int)))) )
         => ( Y3 = X4 ) ) ) ).

% floor_exists1
tff(fact_5662_floor__exists1,axiom,
    ! [X2: 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)),X2))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y3: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),X2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int)))) )
         => ( Y3 = X4 ) ) ) ).

% floor_exists1
tff(fact_5663_tanh__ln__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X2),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5664_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ff),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_5665_take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
      & ( ( N != zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_5666_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_5667_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_5668_round__unique,axiom,
    ! [X2: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),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,X2),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,X2) = Y ) ) ) ).

% round_unique
tff(fact_5669_round__unique,axiom,
    ! [X2: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X2),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(X2),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,X2) = Y ) ) ) ).

% round_unique
tff(fact_5670_sqrt__sum__squares__half__less,axiom,
    ! [X2: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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),X2))
         => ( 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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_5671_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_5672_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_5673_real__sqrt__less__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X2)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% real_sqrt_less_iff
tff(fact_5674_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_5675_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_5676_tanh__real__less__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X2)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% tanh_real_less_iff
tff(fact_5677_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_5678_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_5679_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_5680_real__sqrt__lt__0__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_5681_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_5682_real__sqrt__lt__1__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X2)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_5683_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5684_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5685_tanh__real__neg__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ).

% tanh_real_neg_iff
tff(fact_5686_tanh__real__pos__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% tanh_real_pos_iff
tff(fact_5687_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_5688_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_5689_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)) = 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_5690_take__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_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_5691_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_5692_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(N,A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_5693_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_5694_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_5695_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5696_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_5697_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_5698_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : bit_se1745604003318907178nteger(M,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% take_bit_of_exp
tff(fact_5699_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),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,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),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_5700_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),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),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_of_exp
tff(fact_5701_take__bit__of__2,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(N,numera6620942414471956472nteger(aa_num_num(bit0,one))) = times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5702_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,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_5703_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_5704_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_5705_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_5706_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M: 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,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_5707_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M: 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,M),N))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_5708_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_5709_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: 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),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_5710_real__sqrt__less__mono,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X2)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_5711_round__mono,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X2)),aa_rat_int(archim7778729529865785530nd_rat,Y))) ) ).

% round_mono
tff(fact_5712_real__sqrt__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X2))) ) ).

% real_sqrt_gt_zero
tff(fact_5713_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_5714_not__take__bit__negative,axiom,
    ! [N: nat,K: 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),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_5715_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_5716_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,M),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),M),aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_nat_fun_int_int(bit_ri631733984087533419it_int,M)),A) ).

% signed_take_bit_take_bit
tff(fact_5717_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),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),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_5718_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),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),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_5719_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),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),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_5720_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),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),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_5721_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),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),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_5722_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),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),M))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_5723_tanh__real__lt__1,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X2)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_5724_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_5725_tanh__real__gt__neg1,axiom,
    ! [X2: 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,X2))) ).

% tanh_real_gt_neg1
tff(fact_5726_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_5727_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_5728_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: 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),M)),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_5729_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_5730_real__less__rsqrt,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_5731_take__bit__int__less__exp,axiom,
    ! [N: nat,K: 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),K)),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_5732_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_5733_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_5734_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_5735_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: 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),M)),M))
    <=> 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)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_5736_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_5737_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_5738_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: 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),K)),K))
    <=> 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)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_5739_real__less__lsqrt,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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,X2),aa_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,X2)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_5740_ln__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_5741_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_5742_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_5743_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_5744_ex__le__of__int,axiom,
    ! [X2: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_le_of_int
tff(fact_5745_ex__le__of__int,axiom,
    ! [X2: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_le_of_int
tff(fact_5746_ex__less__of__int,axiom,
    ! [X2: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_less_of_int
tff(fact_5747_ex__less__of__int,axiom,
    ! [X2: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_less_of_int
tff(fact_5748_ex__of__int__less,axiom,
    ! [X2: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X2)) ).

% ex_of_int_less
tff(fact_5749_ex__of__int__less,axiom,
    ! [X2: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X2)) ).

% ex_of_int_less
tff(fact_5750_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5751_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_5752_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_5753_arsinh__real__aux,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_5754_of__int__round__le,axiom,
    ! [X2: 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,X2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),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_5755_of__int__round__le,axiom,
    ! [X2: 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,X2))),aa_rat_rat(plus_plus_rat(X2),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_5756_of__int__round__ge,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),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,X2)))) ).

% of_int_round_ge
tff(fact_5757_of__int__round__ge,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X2),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,X2)))) ).

% of_int_round_ge
tff(fact_5758_of__int__round__gt,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X2),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,X2)))) ).

% of_int_round_gt
tff(fact_5759_of__int__round__gt,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),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,X2)))) ).

% of_int_round_gt
tff(fact_5760_take__bit__int__less__eq,axiom,
    ! [N: nat,K: 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)),K))
     => ( 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),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_5761_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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,K),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),K))) ) ).

% take_bit_int_greater_eq
tff(fact_5762_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
        & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( bit_se1745604003318907178nteger(N,A) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5763_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_5764_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_5765_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),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,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_5766_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_fg(D))) ).

% int_ge_less_than_def
tff(fact_5767_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_fh(D))) ).

% int_ge_less_than2_def
tff(fact_5768_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_ch,set_or1266510415728281911st_int(M,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,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_5769_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_5770_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5771_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5772_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5773_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A2))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_5774_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),set_or1269000886237332187st_nat(M,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,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_5775_insert__absorb2,axiom,
    ! [X2: nat,A2: set_nat] : insert_nat(X2,insert_nat(X2,A2)) = insert_nat(X2,A2) ).

% insert_absorb2
tff(fact_5776_insert__absorb2,axiom,
    ! [X2: int,A2: set_int] : insert_int(X2,insert_int(X2,A2)) = insert_int(X2,A2) ).

% insert_absorb2
tff(fact_5777_insert__absorb2,axiom,
    ! [X2: real,A2: set_real] : insert_real(X2,insert_real(X2,A2)) = insert_real(X2,A2) ).

% insert_absorb2
tff(fact_5778_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),insert_nat(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_nat_bool(member_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_5779_insert__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),insert_VEBT_VEBT(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2)) ) ) ).

% insert_iff
tff(fact_5780_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),insert_int(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_int_bool(member_int(A),A2)) ) ) ).

% insert_iff
tff(fact_5781_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ) ).

% insert_iff
tff(fact_5782_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),insert_real(B,A2)))
    <=> ( ( A = B )
        | pp(aa_set_real_bool(member_real(A),A2)) ) ) ).

% insert_iff
tff(fact_5783_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ pp(aa_set_nat_bool(member_nat(A),B2))
       => ( A = B ) )
     => pp(aa_set_nat_bool(member_nat(A),insert_nat(B,B2))) ) ).

% insertCI
tff(fact_5784_insertCI,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),B2))
       => ( A = B ) )
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),insert_VEBT_VEBT(B,B2))) ) ).

% insertCI
tff(fact_5785_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ pp(aa_set_int_bool(member_int(A),B2))
       => ( A = B ) )
     => pp(aa_set_int_bool(member_int(A),insert_int(B,B2))) ) ).

% insertCI
tff(fact_5786_insertCI,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),B2))
       => ( A = B ) )
     => pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,B2))) ) ).

% insertCI
tff(fact_5787_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ pp(aa_set_real_bool(member_real(A),B2))
       => ( A = B ) )
     => pp(aa_set_real_bool(member_real(A),insert_real(B,B2))) ) ).

% insertCI
tff(fact_5788_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_5789_singletonI,axiom,
    ! [A: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ).

% singletonI
tff(fact_5790_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_5791_singletonI,axiom,
    ! [A: real] : pp(aa_set_real_bool(member_real(A),insert_real(A,bot_bot_set_real))) ).

% singletonI
tff(fact_5792_singletonI,axiom,
    ! [A: nat] : pp(aa_set_nat_bool(member_nat(A),insert_nat(A,bot_bot_set_nat))) ).

% singletonI
tff(fact_5793_singletonI,axiom,
    ! [A: int] : pp(aa_set_int_bool(member_int(A),insert_int(A,bot_bot_set_int))) ).

% singletonI
tff(fact_5794_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,insert_real(A,A2)))
    <=> pp(aa_set_real_bool(finite_finite_real,A2)) ) ).

% finite_insert
tff(fact_5795_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A2)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ).

% finite_insert
tff(fact_5796_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,insert_int(A,A2)))
    <=> pp(aa_set_int_bool(finite_finite_int,A2)) ) ).

% finite_insert
tff(fact_5797_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A2)))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ).

% finite_insert
tff(fact_5798_insert__subset,axiom,
    ! [X2: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X2,A2)),B2))
    <=> ( pp(aa_set_nat_bool(member_nat(X2),B2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ).

% insert_subset
tff(fact_5799_insert__subset,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(X2,A2)),B2))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2)) ) ) ).

% insert_subset
tff(fact_5800_insert__subset,axiom,
    ! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X2,A2)),B2))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(X2),B2))
        & pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% insert_subset
tff(fact_5801_insert__subset,axiom,
    ! [X2: real,A2: set_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X2,A2)),B2))
    <=> ( pp(aa_set_real_bool(member_real(X2),B2))
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ).

% insert_subset
tff(fact_5802_insert__subset,axiom,
    ! [X2: int,A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X2,A2)),B2))
    <=> ( pp(aa_set_int_bool(member_int(X2),B2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ).

% insert_subset
tff(fact_5803_insert__Diff1,axiom,
    ! [X2: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A2),B2) = minus_5127226145743854075T_VEBT(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5804_insert__Diff1,axiom,
    ! [X2: int,B2: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(X2),B2))
     => ( minus_minus_set_int(insert_int(X2,A2),B2) = minus_minus_set_int(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5805_insert__Diff1,axiom,
    ! [X2: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X2),B2))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X2,A2),B2) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5806_insert__Diff1,axiom,
    ! [X2: real,B2: set_real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(X2),B2))
     => ( minus_minus_set_real(insert_real(X2,A2),B2) = minus_minus_set_real(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5807_insert__Diff1,axiom,
    ! [X2: nat,B2: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),B2))
     => ( minus_minus_set_nat(insert_nat(X2,A2),B2) = minus_minus_set_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_5808_Diff__insert0,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
     => ( minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,B2)) = minus_5127226145743854075T_VEBT(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5809_Diff__insert0,axiom,
    ! [X2: int,A2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),A2))
     => ( minus_minus_set_int(A2,insert_int(X2,B2)) = minus_minus_set_int(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5810_Diff__insert0,axiom,
    ! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
     => ( minus_2163939370556025621et_nat(A2,insert_set_nat(X2,B2)) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5811_Diff__insert0,axiom,
    ! [X2: real,A2: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),A2))
     => ( minus_minus_set_real(A2,insert_real(X2,B2)) = minus_minus_set_real(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5812_Diff__insert0,axiom,
    ! [X2: nat,A2: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
     => ( minus_minus_set_nat(A2,insert_nat(X2,B2)) = minus_minus_set_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_5813_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(80)
tff(fact_5814_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_cc,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_5815_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cb),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5816_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_by,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_5817_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fi),A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_5818_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_5819_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_5820_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_5821_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_5822_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_5823_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_fj(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_5824_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_fk,A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_5825_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fl,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_5826_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fm,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_5827_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(aTP_Lamm_fn,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_5828_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_5829_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_5830_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_5831_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_5832_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_5833_sum_Oempty,axiom,
    ! [G: fun_nat_complex] : groups2073611262835488442omplex(G,bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_5834_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_5835_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_5836_sum_Oempty,axiom,
    ! [G: fun_int_complex] : groups3049146728041665814omplex(G,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_5837_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_5838_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 )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),F3))
           => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5839_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 )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),F3))
           => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5840_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 )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),F3))
           => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5841_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5842_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5843_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5844_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5845_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2906978787729119204at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5846_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5847_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5848_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5849_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5850_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups3539618377306564664at_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_5851_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( insert_real(A,A2) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5852_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5853_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int(A,A2) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5854_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5855_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B,bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5856_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5857_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5858_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( set_or1266510415728281911st_int(A,B) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5859_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real(C,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_5860_atLeastAtMost__singleton,axiom,
    ! [A: nat] : set_or1269000886237332187st_nat(A,A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_5861_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_5862_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_5863_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_5864_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_5865_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_5866_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,insert_real(A,B2))))
    <=> pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5867_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,insert_int(A,B2))))
    <=> pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5868_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,insert_complex(A,B2))))
    <=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5869_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,insert_nat(A,B2))))
    <=> pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2))) ) ).

% finite_Diff_insert
tff(fact_5870_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(81)
tff(fact_5871_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_5872_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_fo(A),B),S2) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_fo(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5873_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_fp(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_fp(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5874_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_fq(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_fq(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5875_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_fr(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_fr(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5876_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_fs(A),B),S2) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_fs(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5877_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_ft(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ft(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5878_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_fu(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fu(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5879_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_fv(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fv(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5880_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_fw(A),B),S2) = aa_VEBT_VEBT_rat(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_fw(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5881_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_fx(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fx(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5882_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_fy(A),B),S2) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_fy(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5883_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_fz(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_fz(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5884_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_ga(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ga(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5885_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_gb(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gb(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5886_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_gc(A),B),S2) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_gc(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5887_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_gd(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gd(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5888_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_ge(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ge(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5889_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_gf(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_gf(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5890_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_gg(A),B),S2) = aa_VEBT_VEBT_rat(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_gg(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5891_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_gh(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gh(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5892_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X2)),groups2240296850493347238T_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5893_sum_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups8097168146408367636l_real(G,insert_real(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X2)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5894_sum_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A2))
       => ( groups8778361861064173332t_real(G,insert_int(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5895_sum_Oinsert,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups5808333547571424918x_real(G,insert_complex(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5896_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,X2)),groups136491112297645522BT_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5897_sum_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups1300246762558778688al_rat(G,insert_real(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X2)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5898_sum_Oinsert,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
       => ( groups2906978787729119204at_rat(G,insert_nat(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,X2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5899_sum_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A2))
       => ( groups3906332499630173760nt_rat(G,insert_int(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5900_sum_Oinsert,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups5058264527183730370ex_rat(G,insert_complex(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5901_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X2,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X2)),groups771621172384141258BT_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_5902_subset__Compl__singleton,axiom,
    ! [A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),uminus8041839845116263051T_VEBT(insert_VEBT_VEBT(B,bot_bo8194388402131092736T_VEBT))))
    <=> ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5903_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat))))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5904_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real))))
    <=> ~ pp(aa_set_real_bool(member_real(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5905_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat))))
    <=> ~ pp(aa_set_nat_bool(member_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5906_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int))))
    <=> ~ pp(aa_set_int_bool(member_int(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_5907_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_gi(F),A2) ).

% of_int_sum
tff(fact_5908_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gj(F)),A2) ).

% of_int_sum
tff(fact_5909_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_gk(F),A2) ).

% of_int_sum
tff(fact_5910_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_gl(F),A2) ).

% of_int_sum
tff(fact_5911_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_gm(F),A2) ).

% of_int_sum
tff(fact_5912_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(74)
tff(fact_5913_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(79)
tff(fact_5914_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_5915_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5916_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5917_set__replicate,axiom,
    ! [N: nat,X2: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X2)) = insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_5918_set__replicate,axiom,
    ! [N: nat,X2: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,X2)) = insert_real(X2,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_5919_set__replicate,axiom,
    ! [N: nat,X2: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X2)) = insert_nat(X2,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_5920_set__replicate,axiom,
    ! [N: nat,X2: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X2)) = insert_int(X2,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_5921_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,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)),M))
       => ( groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,set_or1269000886237332187st_nat(M,N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5922_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,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)),M))
       => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5923_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,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)),M))
       => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5924_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,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)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5925_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,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)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5926_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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_5927_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: 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,M)))),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))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5928_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_5929_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_5930_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_5931_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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_5932_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: 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,M)))),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))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5933_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_5934_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_5935_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_5936_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_gn(C),A2) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_gn(C),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_5937_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_go(C),A2) = aa_nat_rat(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_go(C),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_5938_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(C)),A2) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gp(C)),A2) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_5939_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gq(C),D),A2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gq(C),D),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_5940_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_rat,D: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gr(C),D),A2) = aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gr(C),D),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_5941_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gs(C),D)),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gs(C),D)),A2) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_5942_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5943_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5944_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),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,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5945_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5946_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5947_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),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,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5948_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gt),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5949_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N)) = produc4245557441103728435nt_int(aTP_Lamm_gu,unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5950_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N)) = produc6916734918728496179nteger(aTP_Lamm_gv,unique3479559517661332726nteger(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_5951_insert__compr,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,B2) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_gw(A),B2)) ).

% insert_compr
tff(fact_5952_insert__compr,axiom,
    ! [A: real,B2: set_real] : insert_real(A,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_gx(A),B2)) ).

% insert_compr
tff(fact_5953_insert__compr,axiom,
    ! [A: list_nat,B2: set_list_nat] : insert_list_nat(A,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_gy(A),B2)) ).

% insert_compr
tff(fact_5954_insert__compr,axiom,
    ! [A: set_nat,B2: set_set_nat] : insert_set_nat(A,B2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_gz(A),B2)) ).

% insert_compr
tff(fact_5955_insert__compr,axiom,
    ! [A: nat,B2: set_nat] : insert_nat(A,B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_ha(A),B2)) ).

% insert_compr
tff(fact_5956_insert__compr,axiom,
    ! [A: int,B2: set_int] : insert_int(A,B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_hb(A),B2)) ).

% insert_compr
tff(fact_5957_insert__Collect,axiom,
    ! [A: real,P: fun_real_bool] : insert_real(A,collect_real(P)) = collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_hc(A),P)) ).

% insert_Collect
tff(fact_5958_insert__Collect,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] : insert_list_nat(A,collect_list_nat(P)) = collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_hd(A),P)) ).

% insert_Collect
tff(fact_5959_insert__Collect,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] : insert_set_nat(A,collect_set_nat(P)) = collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_he(A),P)) ).

% insert_Collect
tff(fact_5960_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_bool] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_hf(A),P)) ).

% insert_Collect
tff(fact_5961_insert__Collect,axiom,
    ! [A: int,P: fun_int_bool] : insert_int(A,collect_int(P)) = collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_hg(A),P)) ).

% insert_Collect
tff(fact_5962_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ? [B8: set_nat] :
          ( ( A2 = insert_nat(A,B8) )
          & ~ pp(aa_set_nat_bool(member_nat(A),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_5963_mk__disjoint__insert,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
     => ? [B8: set_VEBT_VEBT] :
          ( ( A2 = insert_VEBT_VEBT(A,B8) )
          & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_5964_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ? [B8: set_int] :
          ( ( A2 = insert_int(A,B8) )
          & ~ pp(aa_set_int_bool(member_int(A),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_5965_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ? [B8: set_set_nat] :
          ( ( A2 = insert_set_nat(A,B8) )
          & ~ pp(aa_set_set_nat_bool(member_set_nat(A),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_5966_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ? [B8: set_real] :
          ( ( A2 = insert_real(A,B8) )
          & ~ pp(aa_set_real_bool(member_real(A),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_5967_insert__commute,axiom,
    ! [X2: nat,Y: nat,A2: set_nat] : insert_nat(X2,insert_nat(Y,A2)) = insert_nat(Y,insert_nat(X2,A2)) ).

% insert_commute
tff(fact_5968_insert__commute,axiom,
    ! [X2: int,Y: int,A2: set_int] : insert_int(X2,insert_int(Y,A2)) = insert_int(Y,insert_int(X2,A2)) ).

% insert_commute
tff(fact_5969_insert__commute,axiom,
    ! [X2: real,Y: real,A2: set_real] : insert_real(X2,insert_real(Y,A2)) = insert_real(Y,insert_real(X2,A2)) ).

% insert_commute
tff(fact_5970_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(B),B2))
       => ( ( insert_nat(A,A2) = insert_nat(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_nat] :
                  ( ( A2 = insert_nat(B,C6) )
                  & ~ pp(aa_set_nat_bool(member_nat(B),C6))
                  & ( B2 = insert_nat(A,C6) )
                  & ~ pp(aa_set_nat_bool(member_nat(A),C6)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5971_insert__eq__iff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),B2))
       => ( ( insert_VEBT_VEBT(A,A2) = insert_VEBT_VEBT(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_VEBT_VEBT] :
                  ( ( A2 = insert_VEBT_VEBT(B,C6) )
                  & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),C6))
                  & ( B2 = insert_VEBT_VEBT(A,C6) )
                  & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),C6)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5972_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(A),A2))
     => ( ~ pp(aa_set_int_bool(member_int(B),B2))
       => ( ( insert_int(A,A2) = insert_int(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_int] :
                  ( ( A2 = insert_int(B,C6) )
                  & ~ pp(aa_set_int_bool(member_int(B),C6))
                  & ( B2 = insert_int(A,C6) )
                  & ~ pp(aa_set_int_bool(member_int(A),C6)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5973_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(B),B2))
       => ( ( insert_set_nat(A,A2) = insert_set_nat(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_set_nat] :
                  ( ( A2 = insert_set_nat(B,C6) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(B),C6))
                  & ( B2 = insert_set_nat(A,C6) )
                  & ~ pp(aa_set_set_nat_bool(member_set_nat(A),C6)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5974_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(A),A2))
     => ( ~ pp(aa_set_real_bool(member_real(B),B2))
       => ( ( insert_real(A,A2) = insert_real(B,B2) )
        <=> ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_real] :
                  ( ( A2 = insert_real(B,C6) )
                  & ~ pp(aa_set_real_bool(member_real(B),C6))
                  & ( B2 = insert_real(A,C6) )
                  & ~ pp(aa_set_real_bool(member_real(A),C6)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_5975_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( insert_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5976_insert__absorb,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
     => ( insert_VEBT_VEBT(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5977_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( insert_int(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5978_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( insert_set_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5979_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( insert_real(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_5980_insert__ident,axiom,
    ! [X2: nat,A2: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X2),B2))
       => ( ( insert_nat(X2,A2) = insert_nat(X2,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5981_insert__ident,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
       => ( ( insert_VEBT_VEBT(X2,A2) = insert_VEBT_VEBT(X2,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5982_insert__ident,axiom,
    ! [X2: int,A2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),A2))
     => ( ~ pp(aa_set_int_bool(member_int(X2),B2))
       => ( ( insert_int(X2,A2) = insert_int(X2,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5983_insert__ident,axiom,
    ! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
     => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),B2))
       => ( ( insert_set_nat(X2,A2) = insert_set_nat(X2,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5984_insert__ident,axiom,
    ! [X2: real,A2: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),A2))
     => ( ~ pp(aa_set_real_bool(member_real(X2),B2))
       => ( ( insert_real(X2,A2) = insert_real(X2,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_5985_Set_Oset__insert,axiom,
    ! [X2: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),A2))
     => ~ ! [B8: set_nat] :
            ( ( A2 = insert_nat(X2,B8) )
           => pp(aa_set_nat_bool(member_nat(X2),B8)) ) ) ).

% Set.set_insert
tff(fact_5986_Set_Oset__insert,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
     => ~ ! [B8: set_VEBT_VEBT] :
            ( ( A2 = insert_VEBT_VEBT(X2,B8) )
           => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B8)) ) ) ).

% Set.set_insert
tff(fact_5987_Set_Oset__insert,axiom,
    ! [X2: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(X2),A2))
     => ~ ! [B8: set_int] :
            ( ( A2 = insert_int(X2,B8) )
           => pp(aa_set_int_bool(member_int(X2),B8)) ) ) ).

% Set.set_insert
tff(fact_5988_Set_Oset__insert,axiom,
    ! [X2: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
     => ~ ! [B8: set_set_nat] :
            ( ( A2 = insert_set_nat(X2,B8) )
           => pp(aa_set_set_nat_bool(member_set_nat(X2),B8)) ) ) ).

% Set.set_insert
tff(fact_5989_Set_Oset__insert,axiom,
    ! [X2: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(X2),A2))
     => ~ ! [B8: set_real] :
            ( ( A2 = insert_real(X2,B8) )
           => pp(aa_set_real_bool(member_real(X2),B8)) ) ) ).

% Set.set_insert
tff(fact_5990_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(member_nat(A),B2))
     => pp(aa_set_nat_bool(member_nat(A),insert_nat(B,B2))) ) ).

% insertI2
tff(fact_5991_insertI2,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),B2))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),insert_VEBT_VEBT(B,B2))) ) ).

% insertI2
tff(fact_5992_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( pp(aa_set_int_bool(member_int(A),B2))
     => pp(aa_set_int_bool(member_int(A),insert_int(B,B2))) ) ).

% insertI2
tff(fact_5993_insertI2,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),B2))
     => pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,B2))) ) ).

% insertI2
tff(fact_5994_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( pp(aa_set_real_bool(member_real(A),B2))
     => pp(aa_set_real_bool(member_real(A),insert_real(B,B2))) ) ).

% insertI2
tff(fact_5995_insertI1,axiom,
    ! [A: nat,B2: set_nat] : pp(aa_set_nat_bool(member_nat(A),insert_nat(A,B2))) ).

% insertI1
tff(fact_5996_insertI1,axiom,
    ! [A: vEBT_VEBT,B2: set_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),insert_VEBT_VEBT(A,B2))) ).

% insertI1
tff(fact_5997_insertI1,axiom,
    ! [A: int,B2: set_int] : pp(aa_set_int_bool(member_int(A),insert_int(A,B2))) ).

% insertI1
tff(fact_5998_insertI1,axiom,
    ! [A: set_nat,B2: set_set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(A,B2))) ).

% insertI1
tff(fact_5999_insertI1,axiom,
    ! [A: real,B2: set_real] : pp(aa_set_real_bool(member_real(A),insert_real(A,B2))) ).

% insertI1
tff(fact_6000_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),insert_nat(B,A2)))
     => ( ( A != B )
       => pp(aa_set_nat_bool(member_nat(A),A2)) ) ) ).

% insertE
tff(fact_6001_insertE,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),insert_VEBT_VEBT(B,A2)))
     => ( ( A != B )
       => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2)) ) ) ).

% insertE
tff(fact_6002_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),insert_int(B,A2)))
     => ( ( A != B )
       => pp(aa_set_int_bool(member_int(A),A2)) ) ) ).

% insertE
tff(fact_6003_insertE,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),insert_set_nat(B,A2)))
     => ( ( A != B )
       => pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ) ).

% insertE
tff(fact_6004_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),insert_real(B,A2)))
     => ( ( A != B )
       => pp(aa_set_real_bool(member_real(A),A2)) ) ) ).

% insertE
tff(fact_6005_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X2,A2)) = groups2240296850493347238T_real(G,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X2)),groups2240296850493347238T_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6006_sum_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X2,A2)) = groups8097168146408367636l_real(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X2)),groups8097168146408367636l_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6007_sum_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups8778361861064173332t_real(G,insert_int(X2,A2)) = groups8778361861064173332t_real(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups8778361861064173332t_real(G,insert_int(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X2)),groups8778361861064173332t_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6008_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups5808333547571424918x_real(G,insert_complex(X2,A2)) = groups5808333547571424918x_real(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups5808333547571424918x_real(G,insert_complex(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6009_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X2,A2)) = groups136491112297645522BT_rat(G,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,X2)),groups136491112297645522BT_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6010_sum_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups1300246762558778688al_rat(G,insert_real(X2,A2)) = groups1300246762558778688al_rat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups1300246762558778688al_rat(G,insert_real(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X2)),groups1300246762558778688al_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6011_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( groups2906978787729119204at_rat(G,insert_nat(X2,A2)) = groups2906978787729119204at_rat(G,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( groups2906978787729119204at_rat(G,insert_nat(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,X2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6012_sum_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups3906332499630173760nt_rat(G,insert_int(X2,A2)) = groups3906332499630173760nt_rat(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups3906332499630173760nt_rat(G,insert_int(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6013_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups5058264527183730370ex_rat(G,insert_complex(X2,A2)) = groups5058264527183730370ex_rat(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups5058264527183730370ex_rat(G,insert_complex(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6014_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X2,A2)) = groups771621172384141258BT_nat(G,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X2,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X2)),groups771621172384141258BT_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_6015_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups2073611262835488442omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6016_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VE5569070535936029429omplex,A2: set_VEBT_VEBT] :
      ( ( groups1794756597179926696omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
           => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6017_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups3049146728041665814omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6018_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups5754745047067104278omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6019_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A2: set_VEBT_VEBT] :
      ( ( groups2240296850493347238T_real(G,A2) != zero_zero_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
           => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6020_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6021_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6022_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6023_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_rat,A2: set_VEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
           => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6024_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6025_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( aa_int_int(G,X4) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_6026_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_6027_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_6028_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_real(G,X4) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_6029_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_6030_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_6031_singletonD,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))
     => ( B = A ) ) ).

% singletonD
tff(fact_6032_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_6033_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_6034_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_6035_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_6036_singleton__iff,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_6037_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_6038_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_6039_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_6040_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_6041_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( insert_real(A,insert_real(B,bot_bot_set_real)) = insert_real(C,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6042_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B,bot_bot_set_nat)) = insert_nat(C,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6043_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_6044_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_6045_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_6046_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) != bot_bot_set_int ).

% insert_not_empty
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,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => pp(aa_set_real_bool(finite_finite_real,insert_real(A,A2))) ) ).

% finite.insertI
tff(fact_6051_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_nat_bool(finite_finite_nat,insert_nat(A,A2))) ) ).

% finite.insertI
tff(fact_6052_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => pp(aa_set_int_bool(finite_finite_int,insert_int(A,A2))) ) ).

% finite.insertI
tff(fact_6053_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,insert_complex(A,A2))) ) ).

% finite.insertI
tff(fact_6054_insert__mono,axiom,
    ! [C4: set_nat,D5: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C4),D5))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(A,C4)),insert_nat(A,D5))) ) ).

% insert_mono
tff(fact_6055_insert__mono,axiom,
    ! [C4: set_real,D5: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(C4),D5))
     => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(A,C4)),insert_real(A,D5))) ) ).

% insert_mono
tff(fact_6056_insert__mono,axiom,
    ! [C4: set_int,D5: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C4),D5))
     => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(A,C4)),insert_int(A,D5))) ) ).

% insert_mono
tff(fact_6057_subset__insert,axiom,
    ! [X2: nat,A2: set_nat,B2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X2,B2)))
      <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ).

% subset_insert
tff(fact_6058_subset__insert,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(X2,B2)))
      <=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2)) ) ) ).

% subset_insert
tff(fact_6059_subset__insert,axiom,
    ! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),insert_set_nat(X2,B2)))
      <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% subset_insert
tff(fact_6060_subset__insert,axiom,
    ! [X2: real,A2: set_real,B2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X2,B2)))
      <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ).

% subset_insert
tff(fact_6061_subset__insert,axiom,
    ! [X2: int,A2: set_int,B2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X2,B2)))
      <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ).

% subset_insert
tff(fact_6062_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),insert_nat(A,B2))) ).

% subset_insertI
tff(fact_6063_subset__insertI,axiom,
    ! [B2: set_real,A: real] : pp(aa_set_real_bool(ord_less_eq_set_real(B2),insert_real(A,B2))) ).

% subset_insertI
tff(fact_6064_subset__insertI,axiom,
    ! [B2: set_int,A: int] : pp(aa_set_int_bool(ord_less_eq_set_int(B2),insert_int(A,B2))) ).

% subset_insertI
tff(fact_6065_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(B,B2))) ) ).

% subset_insertI2
tff(fact_6066_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(B,B2))) ) ).

% subset_insertI2
tff(fact_6067_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(B,B2))) ) ).

% subset_insertI2
tff(fact_6068_insert__subsetI,axiom,
    ! [X2: nat,A2: set_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X2),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,insert_nat(X2,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_6069_insert__subsetI,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(X6),A2))
       => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(X2,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_6070_insert__subsetI,axiom,
    ! [X2: set_nat,A2: set_set_nat,X6: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(X6),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(insert_set_nat(X2,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_6071_insert__subsetI,axiom,
    ! [X2: real,A2: set_real,X6: set_real] :
      ( pp(aa_set_real_bool(member_real(X2),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(insert_real(X2,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_6072_insert__subsetI,axiom,
    ! [X2: int,A2: set_int,X6: set_int] :
      ( pp(aa_set_int_bool(member_int(X2),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(insert_int(X2,X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_6073_insert__Diff__if,axiom,
    ! [X2: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
       => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A2),B2) = minus_5127226145743854075T_VEBT(A2,B2) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
       => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A2),B2) = insert_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_6074_insert__Diff__if,axiom,
    ! [X2: int,B2: set_int,A2: set_int] :
      ( ( pp(aa_set_int_bool(member_int(X2),B2))
       => ( minus_minus_set_int(insert_int(X2,A2),B2) = minus_minus_set_int(A2,B2) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X2),B2))
       => ( minus_minus_set_int(insert_int(X2,A2),B2) = insert_int(X2,minus_minus_set_int(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_6075_insert__Diff__if,axiom,
    ! [X2: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(X2),B2))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X2,A2),B2) = minus_2163939370556025621et_nat(A2,B2) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),B2))
       => ( minus_2163939370556025621et_nat(insert_set_nat(X2,A2),B2) = insert_set_nat(X2,minus_2163939370556025621et_nat(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_6076_insert__Diff__if,axiom,
    ! [X2: real,B2: set_real,A2: set_real] :
      ( ( pp(aa_set_real_bool(member_real(X2),B2))
       => ( minus_minus_set_real(insert_real(X2,A2),B2) = minus_minus_set_real(A2,B2) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X2),B2))
       => ( minus_minus_set_real(insert_real(X2,A2),B2) = insert_real(X2,minus_minus_set_real(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_6077_insert__Diff__if,axiom,
    ! [X2: nat,B2: set_nat,A2: set_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X2),B2))
       => ( minus_minus_set_nat(insert_nat(X2,A2),B2) = minus_minus_set_nat(A2,B2) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X2),B2))
       => ( minus_minus_set_nat(insert_nat(X2,A2),B2) = insert_nat(X2,minus_minus_set_nat(A2,B2)) ) ) ) ).

% insert_Diff_if
tff(fact_6078_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_6079_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,K5)),groups136491112297645522BT_rat(G,K5))) ) ).

% sum_mono
tff(fact_6080_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_6081_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_6082_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,K5)),groups771621172384141258BT_nat(G,K5))) ) ).

% sum_mono
tff(fact_6083_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_6084_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_6085_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,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5))) ) ).

% sum_mono
tff(fact_6086_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_VEBT_VEBT_int(F,I3)),aa_VEBT_VEBT_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups769130701875090982BT_int(F,K5)),groups769130701875090982BT_int(G,K5))) ) ).

% sum_mono
tff(fact_6087_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_6088_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hh(G),H),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_6089_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hi(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_6090_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_hj(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_6091_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(G),H)),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),A2)),aa_set_nat_real(groups6591440286371151544t_real(H),A2)) ).

% sum.distrib
tff(fact_6092_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_int,G: fun_VE294566951569886236nt_int,R: fun_VE3303673267288458579t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups769130701875090982BT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_hm(B2),G),R),A2) = groups4538972089207619220nt_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_ho(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6093_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_int,G: fun_real_fun_int_int,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups1932886352136224148al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_hq(B2),G),R),A2) = groups4538972089207619220nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_hs(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6094_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups3539618377306564664at_int(aa_fun3504929462025808522at_int(aa_fun3325927984641084153at_int(aTP_Lamm_hu(B2),G),R),A2) = groups4538972089207619220nt_int(aa_fun4504283173926863078nt_int(aa_fun5512849559334105941nt_int(aTP_Lamm_hw(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6095_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_int,G: fun_co333780937270390912nt_int,R: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups5690904116761175830ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_hy(B2),G),R),A2) = groups4538972089207619220nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_ib(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6096_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,G: fun_VE597728521808591204at_nat,R: fun_VE3763956218935105199t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups771621172384141258BT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_id(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_if(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6097_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_ih(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_ij(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6098_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_il(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_in(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6099_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_ip(B2),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_is(A2),G),R)),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6100_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_complex,G: fun_VE2006399705683203104omplex,R: fun_VE2128610149277289041x_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups1794756597179926696omplex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_iu(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_iw(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6101_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_complex,G: fun_re7208826350674946562omplex,R: fun_re3010880487650781423x_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_iy(B2),G),R),A2) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_ja(A2),G),R),B2) ) ) ) ).

% sum.swap_restrict
tff(fact_6102_sum__diff1__nat,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => ( groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A2)),aa_VEBT_VEBT_nat(F,A)) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => ( groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups771621172384141258BT_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_6103_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: fun_set_nat_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),aa_set_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = groups8294997508430121362at_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_6104_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: fun_real_nat] :
      ( ( pp(aa_set_real_bool(member_real(A),A2))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),aa_real_nat(F,A)) ) )
      & ( ~ pp(aa_set_real_bool(member_real(A),A2))
       => ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1935376822645274424al_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_6105_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: fun_int_nat] :
      ( ( pp(aa_set_int_bool(member_int(A),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)) ) )
      & ( ~ pp(aa_set_int_bool(member_int(A),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A2) ) ) ) ).

% sum_diff1_nat
tff(fact_6106_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: fun_nat_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,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),A2)),aa_nat_nat(F,A)) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) ) ) ) ).

% sum_diff1_nat
tff(fact_6107_Collect__conv__if2,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jb(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jb(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6108_Collect__conv__if2,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jc(P),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jc(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6109_Collect__conv__if2,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jd(P),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jd(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
tff(fact_6110_Collect__conv__if2,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_je(P),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_je(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_6111_Collect__conv__if2,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_jf(P),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_jf(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
tff(fact_6112_Collect__conv__if,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jg(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jg(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
tff(fact_6113_Collect__conv__if,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jh(P),A)) = insert_set_nat(A,bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jh(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_6114_Collect__conv__if,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ji(P),A)) = insert_real(A,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ji(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if
tff(fact_6115_Collect__conv__if,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jj(P),A)) = insert_nat(A,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jj(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_6116_Collect__conv__if,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_jk(P),A)) = insert_int(A,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_jk(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if
tff(fact_6117_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups2240296850493347238T_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X2)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_6118_sum_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6119_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups136491112297645522BT_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,X2)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_6120_sum_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X2)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6121_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups771621172384141258BT_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X2)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_6122_sum_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X2)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6123_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups769130701875090982BT_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,X2)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_6124_sum_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X2)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_6125_sum_Oremove,axiom,
    ! [A2: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups8097168146408367636l_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X2)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_6126_sum_Oremove,axiom,
    ! [A2: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups1300246762558778688al_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X2)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_6127_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,insert_complex(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X2)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6128_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_rat,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,insert_complex(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,X2)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6129_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,insert_complex(X2,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X2)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6130_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5690904116761175830ex_int(G,insert_complex(X2,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X2)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_6131_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_real,X2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,insert_real(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X2)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6132_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_rat,X2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1300246762558778688al_rat(G,insert_real(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,X2)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6133_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,X2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1935376822645274424al_nat(G,insert_real(X2,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X2)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6134_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,X2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1932886352136224148al_int(G,insert_real(X2,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(G,X2)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_6135_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_real,X2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,insert_int(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X2)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_6136_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_rat,X2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,insert_int(X2,A2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,X2)),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_6137_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
         => ( groups2240296850493347238T_real(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2240296850493347238T_real(F,A2)),aa_VEBT_VEBT_real(F,A)) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
         => ( groups2240296850493347238T_real(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups2240296850493347238T_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6138_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),aa_complex_real(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5808333547571424918x_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6139_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(A),A2))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A2)),aa_real_real(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A2))
         => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups8097168146408367636l_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6140_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(A),A2))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),aa_int_real(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A2))
         => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups8778361861064173332t_real(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6141_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
         => ( groups136491112297645522BT_rat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_rat_rat(minus_minus_rat(groups136491112297645522BT_rat(F,A2)),aa_VEBT_VEBT_rat(F,A)) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
         => ( groups136491112297645522BT_rat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups136491112297645522BT_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6142_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),aa_complex_rat(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5058264527183730370ex_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6143_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(F,A2)),aa_real_rat(F,A)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),A2))
         => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1300246762558778688al_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6144_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(A),A2))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),aa_int_rat(F,A)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),A2))
         => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups3906332499630173760nt_rat(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6145_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
         => ( groups769130701875090982BT_int(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups769130701875090982BT_int(F,A2)),aa_VEBT_VEBT_int(F,A)) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
         => ( groups769130701875090982BT_int(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups769130701875090982BT_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6146_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),aa_complex_int(F,A)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
         => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5690904116761175830ex_int(F,A2) ) ) ) ) ).

% sum_diff1
tff(fact_6147_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups2240296850493347238T_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6148_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_6149_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_6150_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_6151_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6152_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_6153_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_6154_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups771621172384141258BT_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_6155_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_6156_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% sum_nonneg
tff(fact_6157_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6158_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6159_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6160_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6161_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6162_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6163_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6164_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_6165_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_6166_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_6167_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_rat,I6: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(F,I6) = groups136491112297645522BT_rat(G,I6) )
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3))) )
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
         => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
           => ( aa_VEBT_VEBT_rat(F,I) = aa_VEBT_VEBT_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6168_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I6: set_real,G: fun_real_rat,I: 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(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6169_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I6: set_nat,G: fun_nat_rat,I: 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(I),I6))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6170_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I6: set_int,G: fun_int_rat,I: 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(I),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6171_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I6: set_complex,G: fun_complex_rat,I: 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(I),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6172_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_nat,I6: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,I: vEBT_VEBT] :
      ( ( groups771621172384141258BT_nat(F,I6) = groups771621172384141258BT_nat(G,I6) )
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3))) )
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
         => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
           => ( aa_VEBT_VEBT_nat(F,I) = aa_VEBT_VEBT_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6173_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I: 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(I),I6))
         => ( pp(aa_set_real_bool(finite_finite_real,I6))
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6174_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I: 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(I),I6))
         => ( pp(aa_set_int_bool(finite_finite_int,I6))
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6175_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I: 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(I),I6))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6176_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_int,I6: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,I: vEBT_VEBT] :
      ( ( groups769130701875090982BT_int(F,I6) = groups769130701875090982BT_int(G,I6) )
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_VEBT_VEBT_int(F,I3)),aa_VEBT_VEBT_int(G,I3))) )
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
         => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
           => ( aa_VEBT_VEBT_int(F,I) = aa_VEBT_VEBT_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6177_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( aa_nat_nat(F,aa_nat_nat(suc,X4)) = aa_nat_nat(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_6178_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A2) = aa_set_nat_real(groups6591440286371151544t_real(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_6179_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_jl(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B,A)),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_jl(A),B),C),S2) = groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6180_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: 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_jm(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,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_jm(A),B),C),S2) = groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6181_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_jn(A),B),C),S2) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(B,A)),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_jn(A),B),C),S2) = groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6182_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: 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_jo(A),B),C),S2) = aa_rat_rat(plus_plus_rat(aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C,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_jo(A),B),C),S2) = groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6183_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_nat,C: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_jp(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B,A)),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_jp(A),B),C),S2) = groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6184_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat,C: 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_jq(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_jq(A),B),C),S2) = groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6185_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_int,C: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_jr(A),B),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(B,A)),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_jr(A),B),C),S2) = groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6186_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int,C: 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_js(A),B),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_js(A),B),C),S2) = groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6187_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C: 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_jt(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B,A)),groups8097168146408367636l_real(C,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_jt(A),B),C),S2) = groups8097168146408367636l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6188_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C: 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_ju(A),B),C),S2) = aa_rat_rat(plus_plus_rat(aa_real_rat(B,A)),groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_ju(A),B),C),S2) = groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% sum.delta_remove
tff(fact_6189_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_6190_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_6191_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_6192_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_6193_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_6194_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_6195_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_6196_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_6197_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_6198_finite__induct,axiom,
    ! [F3: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F3))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X4: vEBT_VEBT,F4: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
             => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),F4))
               => ( pp(aa_set8260762808118238015T_bool(P,F4))
                 => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,F4))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_6199_finite__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(P,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(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_6200_finite__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(P,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(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_6201_finite__induct,axiom,
    ! [F3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(P,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(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_6202_finite__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( pp(aa_set_nat_bool(P,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(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_6203_finite__induct,axiom,
    ! [F3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(P,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(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P,F3)) ) ) ) ).

% finite_induct
tff(fact_6204_finite__ne__induct,axiom,
    ! [F3: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F3))
     => ( ( F3 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT)))
         => ( ! [X4: vEBT_VEBT,F4: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
               => ( ( F4 != bot_bo8194388402131092736T_VEBT )
                 => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),F4))
                   => ( pp(aa_set8260762808118238015T_bool(P,F4))
                     => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,F4))) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6205_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: 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(P,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(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6206_finite__ne__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X4: complex] : pp(aa_set_complex_bool(P,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(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6207_finite__ne__induct,axiom,
    ! [F3: set_real,P: 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(P,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(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6208_finite__ne__induct,axiom,
    ! [F3: set_nat,P: 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(P,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(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6209_finite__ne__induct,axiom,
    ! [F3: set_int,P: 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(P,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(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_ne_induct
tff(fact_6210_infinite__finite__induct,axiom,
    ! [P: fun_se448528567104302406T_bool,A2: set_VEBT_VEBT] :
      ( ! [A5: set_VEBT_VEBT] :
          ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A5))
         => pp(aa_set8260762808118238015T_bool(P,A5)) )
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X4: vEBT_VEBT,F4: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
             => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),F4))
               => ( pp(aa_set8260762808118238015T_bool(P,F4))
                 => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,F4))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_6211_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_bool,A2: set_set_nat] :
      ( ! [A5: set_set_nat] :
          ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A5))
         => pp(aa_set_set_nat_bool(P,A5)) )
     => ( pp(aa_set_set_nat_bool(P,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(P,F4))
                 => pp(aa_set_set_nat_bool(P,insert_set_nat(X4,F4))) ) ) )
         => pp(aa_set_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_6212_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_bool,A2: set_complex] :
      ( ! [A5: set_complex] :
          ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
         => pp(aa_set_complex_bool(P,A5)) )
     => ( pp(aa_set_complex_bool(P,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(P,F4))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,F4))) ) ) )
         => pp(aa_set_complex_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_6213_infinite__finite__induct,axiom,
    ! [P: fun_set_real_bool,A2: set_real] :
      ( ! [A5: set_real] :
          ( ~ pp(aa_set_real_bool(finite_finite_real,A5))
         => pp(aa_set_real_bool(P,A5)) )
     => ( pp(aa_set_real_bool(P,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(P,F4))
                 => pp(aa_set_real_bool(P,insert_real(X4,F4))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_6214_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_bool,A2: set_nat] :
      ( ! [A5: set_nat] :
          ( ~ pp(aa_set_nat_bool(finite_finite_nat,A5))
         => pp(aa_set_nat_bool(P,A5)) )
     => ( pp(aa_set_nat_bool(P,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(P,F4))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,F4))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_6215_infinite__finite__induct,axiom,
    ! [P: fun_set_int_bool,A2: set_int] :
      ( ! [A5: set_int] :
          ( ~ pp(aa_set_int_bool(finite_finite_int,A5))
         => pp(aa_set_int_bool(P,A5)) )
     => ( pp(aa_set_int_bool(P,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(P,F4))
                 => pp(aa_set_int_bool(P,insert_int(X4,F4))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_6216_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_6217_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_6218_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_6219_subset__singletonD,axiom,
    ! [A2: set_real,X2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X2,bot_bot_set_real)))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(X2,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_6220_subset__singletonD,axiom,
    ! [A2: set_nat,X2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X2,bot_bot_set_nat)))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(X2,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_6221_subset__singletonD,axiom,
    ! [A2: set_int,X2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X2,bot_bot_set_int)))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(X2,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_6222_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( set_or1269000886237332187st_nat(A,B) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_6223_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( set_or1266510415728281911st_int(A,B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_6224_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_6225_Diff__insert__absorb,axiom,
    ! [X2: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X2,A2),insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6226_Diff__insert__absorb,axiom,
    ! [X2: set_nat,A2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
     => ( minus_2163939370556025621et_nat(insert_set_nat(X2,A2),insert_set_nat(X2,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6227_Diff__insert__absorb,axiom,
    ! [X2: real,A2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X2),A2))
     => ( minus_minus_set_real(insert_real(X2,A2),insert_real(X2,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6228_Diff__insert__absorb,axiom,
    ! [X2: int,A2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X2),A2))
     => ( minus_minus_set_int(insert_int(X2,A2),insert_int(X2,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6229_Diff__insert__absorb,axiom,
    ! [X2: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
     => ( minus_minus_set_nat(insert_nat(X2,A2),insert_nat(X2,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_6230_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B2) ).

% Diff_insert2
tff(fact_6231_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B2) ).

% Diff_insert2
tff(fact_6232_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat)),B2) ).

% Diff_insert2
tff(fact_6233_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
     => ( insert_VEBT_VEBT(A,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = A2 ) ) ).

% insert_Diff
tff(fact_6234_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_6235_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_6236_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_6237_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_6238_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,B2),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_6239_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_6240_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,B2),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_6241_subset__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,X2: vEBT_VEBT,C4: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B2,insert_VEBT_VEBT(X2,C4))))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B2,C4)))
        & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2)) ) ) ).

% subset_Diff_insert
tff(fact_6242_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X2: set_nat,C4: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,insert_set_nat(X2,C4))))
    <=> ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,C4)))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(X2),A2)) ) ) ).

% subset_Diff_insert
tff(fact_6243_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,X2: real,C4: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B2,insert_real(X2,C4))))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B2,C4)))
        & ~ pp(aa_set_real_bool(member_real(X2),A2)) ) ) ).

% subset_Diff_insert
tff(fact_6244_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X2: nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),minus_minus_set_nat(B2,insert_nat(X2,C4))))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),minus_minus_set_nat(B2,C4)))
        & ~ pp(aa_set_nat_bool(member_nat(X2),A2)) ) ) ).

% subset_Diff_insert
tff(fact_6245_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,X2: int,C4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B2,insert_int(X2,C4))))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B2,C4)))
        & ~ pp(aa_set_int_bool(member_int(X2),A2)) ) ) ).

% subset_Diff_insert
tff(fact_6246_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( groups1794756597179926696omplex(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(A2),P))) = groups1794756597179926696omplex(aa_fun6792494735270171281omplex(aTP_Lamm_jv(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6247_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(A2),P))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_jw(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6248_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bf(A2),P))) = groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_jx(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6249_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(A2),P))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_jy(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6250_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(A2),P))) = groups2240296850493347238T_real(aa_fun2072160908118572431T_real(aTP_Lamm_jz(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6251_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(A2),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_ka(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6252_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(A2),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_kb(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6253_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kc(A2),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_kd(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6254_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( groups136491112297645522BT_rat(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(A2),P))) = groups136491112297645522BT_rat(aa_fun3193099451372820155BT_rat(aTP_Lamm_ke(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6255_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(A2),P))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_kf(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6256_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),A2))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
       => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),groups2240296850493347238T_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6257_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(I,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,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6258_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A2,insert_real(I,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,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6259_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A2,insert_int(I,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,A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6260_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),A2))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
       => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6261_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(member_complex(I),A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A2,insert_complex(I,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,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6262_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A2,insert_real(I,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,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6263_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A2,insert_int(I,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,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6264_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(A2,insert_nat(I,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,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6265_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),A2))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4))) )
       => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I)),groups771621172384141258BT_nat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_6266_sum__subtractf__nat,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(G,X4)),aa_VEBT_VEBT_nat(F,X4))) )
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_kg(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6267_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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_kh(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6268_sum__subtractf__nat,axiom,
    ! [A2: 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),A2))
         => 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_ki(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),groups8294997508430121362at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6269_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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_kj(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6270_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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_kk(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6271_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kl(G)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6272_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_km(G)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6273_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kn(G),K)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6274_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ko(G),K)),set_or1269000886237332187st_nat(M,N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6275_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_real,I: 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(I,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_6276_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_real,I: 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(I,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_6277_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_int_real,I: 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(I,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_6278_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_complex_real,I: 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(I,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_6279_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_rat,I: 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(I,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_6280_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_rat,I: 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(I,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_6281_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: fun_complex_rat,I: 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(I,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_6282_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_nat_rat,I: 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(I,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_6283_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_rat,I: 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(I,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_6284_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_rat,I: 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(I,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_6285_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
       => ( ( groups2240296850493347238T_real(F,A2) = zero_zero_real )
        <=> ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
             => ( aa_VEBT_VEBT_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6286_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A2))
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6287_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6288_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A2))
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6289_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
       => ( ( groups136491112297645522BT_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
             => ( aa_VEBT_VEBT_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6290_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A2))
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6291_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A2))
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6292_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6293_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A2))
             => ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6294_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4))) )
       => ( ( groups771621172384141258BT_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),A2))
             => ( aa_VEBT_VEBT_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6295_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
       => ( ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X)),aa_int_real(G,X))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6296_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
       => ( ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6297_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
       => ( ? [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6298_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
       => ( ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(G,X))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6299_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
       => ( ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6300_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
       => ( ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(G,X))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6301_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
       => ( ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6302_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,X4))) )
       => ( ? [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X)),aa_nat_int(G,X))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6303_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,X4))) )
       => ( ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,X)),aa_complex_int(G,X))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6304_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(G,X4))) )
       => ( ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(F,X)),aa_int_int(G,X))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6305_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))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(plus_plus_complex(X1),Y1)),aa_complex_complex(plus_plus_complex(X23),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_6306_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))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(plus_plus_complex(X1),Y1)),aa_complex_complex(plus_plus_complex(X23),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_6307_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))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),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_6308_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))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),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_6309_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))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),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_6310_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))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),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_6311_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))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),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_6312_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))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),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_6313_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))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),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_6314_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))
     => ( ! [X1: int,Y1: int,X23: int,Y23: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R,X1),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R,Y1),Y23)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),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,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_6315_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_VEBT_VEBT_real(F,X4)),aa_VEBT_VEBT_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6316_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => 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,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6317_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => 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,A2)),groups8097168146408367636l_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6318_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => 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,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6319_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_VEBT_VEBT_rat(F,X4)),aa_VEBT_VEBT_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6320_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => 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,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6321_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => 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,A2)),groups1300246762558778688al_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6322_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => 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,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6323_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => 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,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6324_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_VEBT_VEBT_nat(F,X4)),aa_VEBT_VEBT_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6325_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T3: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T3,T4))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),T4))
                       => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups1794756597179926696omplex(G,S2) = groups1794756597179926696omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6326_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_real,S2: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T3: set_real,G: fun_VE5569070535936029429omplex,H: fun_real_complex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(J,A3)),minus_minus_set_real(T3,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_real_complex(H,aa_VEBT_VEBT_real(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups1794756597179926696omplex(G,S2) = groups5754745047067104278omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6327_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_VEBT_VEBT,S2: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_real_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T3,T4))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                   => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(I,B3)),minus_minus_set_real(S2,S4))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S4))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),T4))
                       => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_real_VEBT_VEBT(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups1794756597179926696omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6328_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T3,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S4))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S4))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( 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,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6329_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_int,S2: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T3: set_int,G: fun_VE5569070535936029429omplex,H: fun_int_complex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(J,A3)),minus_minus_set_int(T3,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_int_complex(H,aa_VEBT_VEBT_int(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups1794756597179926696omplex(G,S2) = groups3049146728041665814omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6330_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T3: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T3,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S4))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S4))
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( 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,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6331_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_VEBT_VEBT,S2: set_int,I: fun_VEBT_VEBT_int,J: fun_int_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_int_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
             => ( aa_VEBT_VEBT_int(I,aa_int_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T3,T4))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                 => ( aa_int_VEBT_VEBT(J,aa_VEBT_VEBT_int(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                   => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),T4))
                       => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_int_VEBT_VEBT(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups1794756597179926696omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6332_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T3: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T3,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( 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,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6333_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T3: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T3,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( 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,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6334_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T3,T4))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),T4))
                       => ( aa_VEBT_VEBT_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_VEBT_VEBT_real(H,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_real(G,A3) ) )
                     => ( groups2240296850493347238T_real(G,S2) = groups2240296850493347238T_real(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6335_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6336_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6337_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6338_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,N) )
     => ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) ) ) ).

% sum_SucD
tff(fact_6339_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6340_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6341_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6342_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_6343_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_6344_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_6345_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_6346_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_6347_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_6348_numeral__Bit1,axiom,
    ! [N: num] : numera6620942414471956472nteger(aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(N)),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_6349_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X4: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
             => ( ! [Y3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y3)),aa_VEBT_VEBT_rat(F,X4))) )
               => ( pp(aa_set8260762808118238015T_bool(P,S5))
                 => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,S5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6350_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y3)),aa_complex_rat(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6351_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S5))
             => ( ! [Y3: real] :
                    ( pp(aa_set_real_bool(member_real(Y3),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y3)),aa_real_rat(F,X4))) )
               => ( pp(aa_set_real_bool(P,S5))
                 => pp(aa_set_real_bool(P,insert_real(X4,S5))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6352_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S5))
             => ( ! [Y3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y3),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y3)),aa_nat_rat(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S5))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S5))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6353_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S5))
             => ( ! [Y3: int] :
                    ( pp(aa_set_int_bool(member_int(Y3),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y3)),aa_int_rat(F,X4))) )
               => ( pp(aa_set_int_bool(P,S5))
                 => pp(aa_set_int_bool(P,insert_int(X4,S5))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6354_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_num] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X4: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
             => ( ! [Y3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y3)),aa_VEBT_VEBT_num(F,X4))) )
               => ( pp(aa_set8260762808118238015T_bool(P,S5))
                 => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,S5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6355_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y3),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y3)),aa_complex_num(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6356_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S5))
             => ( ! [Y3: real] :
                    ( pp(aa_set_real_bool(member_real(Y3),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,Y3)),aa_real_num(F,X4))) )
               => ( pp(aa_set_real_bool(P,S5))
                 => pp(aa_set_real_bool(P,insert_real(X4,S5))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6357_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S5))
             => ( ! [Y3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y3),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,Y3)),aa_nat_num(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S5))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S5))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6358_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S5))
             => ( ! [Y3: int] :
                    ( pp(aa_set_int_bool(member_int(Y3),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,Y3)),aa_int_num(F,X4))) )
               => ( pp(aa_set_int_bool(P,S5))
                 => pp(aa_set_int_bool(P,insert_int(X4,S5))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_6359_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A5))
             => ( ! [X: real] :
                    ( pp(aa_set_real_bool(member_real(X),A5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X)) )
               => ( pp(aa_set_real_bool(P,A5))
                 => pp(aa_set_real_bool(P,insert_real(B3,A5))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6360_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A5: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A5))
             => ( ! [X: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X),A5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),X)) )
               => ( pp(aa_set_rat_bool(P,A5))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A5))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6361_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A5: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A5))
             => ( ! [X: num] :
                    ( pp(aa_set_num_bool(member_num(X),A5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),X)) )
               => ( pp(aa_set_num_bool(P,A5))
                 => pp(aa_set_num_bool(P,insert_num(B3,A5))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6362_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A5))
             => ( ! [X: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X),A5))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),X)) )
               => ( pp(aa_set_nat_bool(P,A5))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A5))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6363_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A5))
             => ( ! [X: int] :
                    ( pp(aa_set_int_bool(member_int(X),A5))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),X)) )
               => ( pp(aa_set_int_bool(P,A5))
                 => pp(aa_set_int_bool(P,insert_int(B3,A5))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_6364_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A5))
             => ( ! [X: real] :
                    ( pp(aa_set_real_bool(member_real(X),A5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B3)) )
               => ( pp(aa_set_real_bool(P,A5))
                 => pp(aa_set_real_bool(P,insert_real(B3,A5))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6365_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A5: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A5))
             => ( ! [X: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X),A5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),B3)) )
               => ( pp(aa_set_rat_bool(P,A5))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A5))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6366_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A5: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A5))
             => ( ! [X: num] :
                    ( pp(aa_set_num_bool(member_num(X),A5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),B3)) )
               => ( pp(aa_set_num_bool(P,A5))
                 => pp(aa_set_num_bool(P,insert_num(B3,A5))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6367_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A5))
             => ( ! [X: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X),A5))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),B3)) )
               => ( pp(aa_set_nat_bool(P,A5))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A5))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6368_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A5))
             => ( ! [X: int] :
                    ( pp(aa_set_int_bool(member_int(X),A5))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),B3)) )
               => ( pp(aa_set_int_bool(P,A5))
                 => pp(aa_set_int_bool(P,insert_int(B3,A5))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_6369_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_6370_finite__subset__induct_H,axiom,
    ! [F3: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F3),A2))
       => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
         => ( ! [A3: vEBT_VEBT,F4: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
               => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
                 => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F4),A2))
                   => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),F4))
                     => ( pp(aa_set8260762808118238015T_bool(P,F4))
                       => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(A3,F4))) ) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6371_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A2))
       => ( pp(aa_set_set_nat_bool(P,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),A2))
                 => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F4),A2))
                   => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                     => ( pp(aa_set_set_nat_bool(P,F4))
                       => pp(aa_set_set_nat_bool(P,insert_set_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6372_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F4),A2))
                   => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                     => ( pp(aa_set_complex_bool(P,F4))
                       => pp(aa_set_complex_bool(P,insert_complex(A3,F4))) ) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6373_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A2))
       => ( pp(aa_set_real_bool(P,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),A2))
                 => ( pp(aa_set_real_bool(ord_less_eq_set_real(F4),A2))
                   => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                     => ( pp(aa_set_real_bool(P,F4))
                       => pp(aa_set_real_bool(P,insert_real(A3,F4))) ) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6374_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: 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),A2))
       => ( pp(aa_set_nat_bool(P,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),A2))
                 => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F4),A2))
                   => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                     => ( pp(aa_set_nat_bool(P,F4))
                       => pp(aa_set_nat_bool(P,insert_nat(A3,F4))) ) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6375_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A2))
       => ( pp(aa_set_int_bool(P,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),A2))
                 => ( pp(aa_set_int_bool(ord_less_eq_set_int(F4),A2))
                   => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                     => ( pp(aa_set_int_bool(P,F4))
                       => pp(aa_set_int_bool(P,insert_int(A3,F4))) ) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_6376_finite__subset__induct,axiom,
    ! [F3: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F3),A2))
       => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
         => ( ! [A3: vEBT_VEBT,F4: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
               => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
                 => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),F4))
                   => ( pp(aa_set8260762808118238015T_bool(P,F4))
                     => pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(A3,F4))) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6377_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F3))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F3),A2))
       => ( pp(aa_set_set_nat_bool(P,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),A2))
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F4))
                   => ( pp(aa_set_set_nat_bool(P,F4))
                     => pp(aa_set_set_nat_bool(P,insert_set_nat(A3,F4))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6378_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F3),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F4: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( ~ pp(aa_set_complex_bool(member_complex(A3),F4))
                   => ( pp(aa_set_complex_bool(P,F4))
                     => pp(aa_set_complex_bool(P,insert_complex(A3,F4))) ) ) ) )
           => pp(aa_set_complex_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6379_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F3),A2))
       => ( pp(aa_set_real_bool(P,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),A2))
                 => ( ~ pp(aa_set_real_bool(member_real(A3),F4))
                   => ( pp(aa_set_real_bool(P,F4))
                     => pp(aa_set_real_bool(P,insert_real(A3,F4))) ) ) ) )
           => pp(aa_set_real_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6380_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: 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),A2))
       => ( pp(aa_set_nat_bool(P,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),A2))
                 => ( ~ pp(aa_set_nat_bool(member_nat(A3),F4))
                   => ( pp(aa_set_nat_bool(P,F4))
                     => pp(aa_set_nat_bool(P,insert_nat(A3,F4))) ) ) ) )
           => pp(aa_set_nat_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6381_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F3),A2))
       => ( pp(aa_set_int_bool(P,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),A2))
                 => ( ~ pp(aa_set_int_bool(member_int(A3),F4))
                   => ( pp(aa_set_int_bool(P,F4))
                     => pp(aa_set_int_bool(P,insert_int(A3,F4))) ) ) ) )
           => pp(aa_set_int_bool(P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_6382_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,B2: real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
       => ( ( groups2240296850493347238T_real(F,S) = B2 )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6383_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B2: real,I: 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) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6384_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B2: real,I: 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) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6385_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B2: real,I: 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) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6386_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,B2: rat,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
       => ( ( groups136491112297645522BT_rat(F,S) = B2 )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6387_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B2: rat,I: 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) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6388_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B2: rat,I: 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) = B2 )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6389_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B2: rat,I: 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) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6390_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B2: rat,I: 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) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6391_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,B2: nat,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3))) )
       => ( ( groups771621172384141258BT_nat(F,S) = B2 )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6392_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
       => ( ( groups2240296850493347238T_real(F,S) = zero_zero_real )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => ( aa_VEBT_VEBT_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6393_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: 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(I),S))
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6394_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: 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(I),S))
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6395_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: 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(I),S))
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6396_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
       => ( ( groups136491112297645522BT_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => ( aa_VEBT_VEBT_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6397_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I: 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(I),S))
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6398_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: 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(I),S))
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6399_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: 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(I),S))
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6400_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: 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(I),S))
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6401_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3))) )
       => ( ( groups771621172384141258BT_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),S))
           => ( aa_VEBT_VEBT_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6402_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_6403_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_6404_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_6405_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_6406_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_bool,A2: set_complex] :
      ( pp(aa_set_complex_bool(X6,A2))
     => ( ! [A5: set_complex] :
            ( pp(aa_set_complex_bool(X6,A5))
           => ? [X: complex] :
                ( pp(aa_set_complex_bool(member_complex(X),A5))
                & ( pp(aa_set_complex_bool(X6,minus_811609699411566653omplex(A5,insert_complex(X,bot_bot_set_complex))))
                  | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A5,insert_complex(X,bot_bot_set_complex)))) ) ) )
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% infinite_coinduct
tff(fact_6407_infinite__coinduct,axiom,
    ! [X6: fun_set_real_bool,A2: set_real] :
      ( pp(aa_set_real_bool(X6,A2))
     => ( ! [A5: set_real] :
            ( pp(aa_set_real_bool(X6,A5))
           => ? [X: real] :
                ( pp(aa_set_real_bool(member_real(X),A5))
                & ( pp(aa_set_real_bool(X6,minus_minus_set_real(A5,insert_real(X,bot_bot_set_real))))
                  | ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A5,insert_real(X,bot_bot_set_real)))) ) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,A2)) ) ) ).

% infinite_coinduct
tff(fact_6408_infinite__coinduct,axiom,
    ! [X6: fun_set_int_bool,A2: set_int] :
      ( pp(aa_set_int_bool(X6,A2))
     => ( ! [A5: set_int] :
            ( pp(aa_set_int_bool(X6,A5))
           => ? [X: int] :
                ( pp(aa_set_int_bool(member_int(X),A5))
                & ( pp(aa_set_int_bool(X6,minus_minus_set_int(A5,insert_int(X,bot_bot_set_int))))
                  | ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A5,insert_int(X,bot_bot_set_int)))) ) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% infinite_coinduct
tff(fact_6409_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_bool,A2: set_nat] :
      ( pp(aa_set_nat_bool(X6,A2))
     => ( ! [A5: set_nat] :
            ( pp(aa_set_nat_bool(X6,A5))
           => ? [X: nat] :
                ( pp(aa_set_nat_bool(member_nat(X),A5))
                & ( pp(aa_set_nat_bool(X6,minus_minus_set_nat(A5,insert_nat(X,bot_bot_set_nat))))
                  | ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A5,insert_nat(X,bot_bot_set_nat)))) ) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% infinite_coinduct
tff(fact_6410_finite__empty__induct,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(P,A2))
       => ( ! [A3: vEBT_VEBT,A5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A5))
             => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A5))
               => ( pp(aa_set8260762808118238015T_bool(P,A5))
                 => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(A3,bot_bo8194388402131092736T_VEBT)))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT)) ) ) ) ).

% finite_empty_induct
tff(fact_6411_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(P,A2))
       => ( ! [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(P,A5))
                 => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A5,insert_set_nat(A3,bot_bot_set_set_nat)))) ) ) )
         => pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6412_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(P,A2))
       => ( ! [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(P,A5))
                 => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A5,insert_complex(A3,bot_bot_set_complex)))) ) ) )
         => pp(aa_set_complex_bool(P,bot_bot_set_complex)) ) ) ) ).

% finite_empty_induct
tff(fact_6413_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,A2))
       => ( ! [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(P,A5))
                 => pp(aa_set_real_bool(P,minus_minus_set_real(A5,insert_real(A3,bot_bot_set_real)))) ) ) )
         => pp(aa_set_real_bool(P,bot_bot_set_real)) ) ) ) ).

% finite_empty_induct
tff(fact_6414_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,A2))
       => ( ! [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(P,A5))
                 => pp(aa_set_int_bool(P,minus_minus_set_int(A5,insert_int(A3,bot_bot_set_int)))) ) ) )
         => pp(aa_set_int_bool(P,bot_bot_set_int)) ) ) ) ).

% finite_empty_induct
tff(fact_6415_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,A2))
       => ( ! [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(P,A5))
                 => pp(aa_set_nat_bool(P,minus_minus_set_nat(A5,insert_nat(A3,bot_bot_set_nat)))) ) ) )
         => pp(aa_set_nat_bool(P,bot_bot_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_6416_Diff__single__insert,axiom,
    ! [A2: set_real,X2: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real))),B2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X2,B2))) ) ).

% Diff_single_insert
tff(fact_6417_Diff__single__insert,axiom,
    ! [A2: set_nat,X2: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X2,bot_bot_set_nat))),B2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X2,B2))) ) ).

% Diff_single_insert
tff(fact_6418_Diff__single__insert,axiom,
    ! [A2: set_int,X2: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int))),B2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X2,B2))) ) ).

% Diff_single_insert
tff(fact_6419_subset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(X2,B2)))
    <=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT))),B2)) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6420_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),insert_set_nat(X2,B2)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X2,bot_bot_set_set_nat))),B2)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6421_subset__insert__iff,axiom,
    ! [A2: set_real,X2: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X2,B2)))
    <=> ( ( pp(aa_set_real_bool(member_real(X2),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real))),B2)) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6422_subset__insert__iff,axiom,
    ! [A2: set_nat,X2: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),insert_nat(X2,B2)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,minus_minus_set_nat(A2,insert_nat(X2,bot_bot_set_nat))),B2)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6423_subset__insert__iff,axiom,
    ! [A2: set_int,X2: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X2,B2)))
    <=> ( ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int))),B2)) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ) ).

% subset_insert_iff
tff(fact_6424_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% atLeast0_atMost_Suc
tff(fact_6425_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( set_or1269000886237332187st_nat(M,N) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_6426_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_6427_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = set_or1269000886237332187st_nat(M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_6428_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kp(G)))) = groups5754745047067104278omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6429_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kq(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6430_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kr(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6431_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ks(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6432_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kt(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6433_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ku(G)))) = groups1300246762558778688al_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6434_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kv(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6435_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kw(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6436_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kx(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6437_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ky(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6438_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X2: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X2))),insert_real(X2,set_real2(Xs)))) ).

% set_update_subset_insert
tff(fact_6439_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,X2))),insert_nat(X2,set_nat2(Xs)))) ).

% set_update_subset_insert
tff(fact_6440_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X2))),insert_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs)))) ).

% set_update_subset_insert
tff(fact_6441_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X2))),insert_int(X2,set_int2(Xs)))) ).

% set_update_subset_insert
tff(fact_6442_Compl__insert,axiom,
    ! [X2: real,A2: set_real] : uminus612125837232591019t_real(insert_real(X2,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(X2,bot_bot_set_real)) ).

% Compl_insert
tff(fact_6443_Compl__insert,axiom,
    ! [X2: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(X2,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(X2,bot_bot_set_int)) ).

% Compl_insert
tff(fact_6444_Compl__insert,axiom,
    ! [X2: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(X2,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat(X2,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_6445_sum__power__add,axiom,
    ! [X2: complex,M: nat,I6: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_kz(X2),M),I6) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X2),M)),groups2073611262835488442omplex(power_power_complex(X2),I6)) ).

% sum_power_add
tff(fact_6446_sum__power__add,axiom,
    ! [X2: rat,M: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_la(X2),M),I6) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),M)),groups2906978787729119204at_rat(power_power_rat(X2),I6)) ).

% sum_power_add
tff(fact_6447_sum__power__add,axiom,
    ! [X2: int,M: nat,I6: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_lb(X2),M),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X2),M)),groups3539618377306564664at_int(power_power_int(X2),I6)) ).

% sum_power_add
tff(fact_6448_sum__power__add,axiom,
    ! [X2: real,M: nat,I6: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lc(X2),M)),I6) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X2),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),I6)) ).

% sum_power_add
tff(fact_6449_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(N,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ld(G),N),M)),set_or1269000886237332187st_nat(N,M)) ).

% sum.atLeastAtMost_rev
tff(fact_6450_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(N,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_le(G),N),M)),set_or1269000886237332187st_nat(N,M)) ).

% sum.atLeastAtMost_rev
tff(fact_6451_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_6452_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_6453_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_6454_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_6455_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_6456_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_6457_numeral__code_I3_J,axiom,
    ! [N: num] : numera6620942414471956472nteger(aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(N)),one_one_Code_integer) ).

% numeral_code(3)
tff(fact_6458_sum__pos2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_6459_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
         => ( ! [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_6460_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
         => ( ! [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_6461_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
         => ( ! [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_6462_sum__pos2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_6463_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I)))
         => ( ! [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_6464_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I)))
         => ( ! [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_6465_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I)))
         => ( ! [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_6466_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I)))
         => ( ! [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_6467_sum__pos2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups771621172384141258BT_nat(F,I6))) ) ) ) ) ).

% sum_pos2
tff(fact_6468_sum__pos,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I6))) ) ) ) ).

% sum_pos
tff(fact_6469_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_6470_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_6471_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_6472_sum__pos,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_6473_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_6474_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_6475_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_6476_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_6477_sum__pos,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups771621172384141258BT_nat(F,I6))) ) ) ) ).

% sum_pos
tff(fact_6478_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_complex(G,X4) = aa_VEBT_VEBT_complex(H,X4) ) )
           => ( groups1794756597179926696omplex(G,T3) = groups1794756597179926696omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6479_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6480_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2240296850493347238T_real(G,T3) = groups2240296850493347238T_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6481_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6482_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6483_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups136491112297645522BT_rat(G,T3) = groups136491112297645522BT_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6484_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6485_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6486_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_nat(G,X4) = aa_VEBT_VEBT_nat(H,X4) ) )
           => ( groups771621172384141258BT_nat(G,T3) = groups771621172384141258BT_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6487_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6488_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VE5569070535936029429omplex,G: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_complex(G,X4) = aa_VEBT_VEBT_complex(H,X4) ) )
           => ( groups1794756597179926696omplex(G,S2) = groups1794756597179926696omplex(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6489_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6490_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2240296850493347238T_real(G,S2) = groups2240296850493347238T_real(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6491_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6492_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6493_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups136491112297645522BT_rat(G,S2) = groups136491112297645522BT_rat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6494_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6495_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6496_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_nat(G,X4) = aa_VEBT_VEBT_nat(H,X4) ) )
           => ( groups771621172384141258BT_nat(G,S2) = groups771621172384141258BT_nat(H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6497_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6498_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T3) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6499_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T3) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6500_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T3) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6501_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T3) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6502_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T3) = groups2073611262835488442omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6503_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T3) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6504_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T3) = groups3539618377306564664at_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6505_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T3) = groups3049146728041665814omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6506_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T3) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6507_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T3) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6508_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6509_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6510_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6511_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6512_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S2) = groups2073611262835488442omplex(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6513_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6514_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S2) = groups3539618377306564664at_int(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6515_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6516_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6517_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T3) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6518_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,C4) = groups1794756597179926696omplex(H,C4) )
               => ( groups1794756597179926696omplex(G,A2) = groups1794756597179926696omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6519_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6520_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_real(H,B3) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) )
               => ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6521_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6522_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6523_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) )
               => ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6524_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6525_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6526_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups771621172384141258BT_nat(G,C4) = groups771621172384141258BT_nat(H,C4) )
               => ( groups771621172384141258BT_nat(G,A2) = groups771621172384141258BT_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6527_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6528_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,A2) = groups1794756597179926696omplex(H,B2) )
              <=> ( groups1794756597179926696omplex(G,C4) = groups1794756597179926696omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6529_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) )
              <=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6530_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_real(H,B3) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B2) )
              <=> ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6531_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6532_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6533_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B2) )
              <=> ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6534_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6535_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6536_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups771621172384141258BT_nat(G,A2) = groups771621172384141258BT_nat(H,B2) )
              <=> ( groups771621172384141258BT_nat(G,C4) = groups771621172384141258BT_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6537_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6538_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6539_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6540_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6541_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6542_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups2906978787729119204at_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2))),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6543_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6544_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6545_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6546_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6547_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups4538972089207619220nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6548_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6549_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,B2)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6550_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6551_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B2)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6552_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6553_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6554_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B2)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6555_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6556_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,B2)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(F,B2)) ) ) ) ).

% sum_diff
tff(fact_6557_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(F),B2)) ) ) ) ).

% sum_diff
tff(fact_6558_sum__diff__nat,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_6559_sum__diff__nat,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ).

% sum_diff_nat
tff(fact_6560_sum__diff__nat,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_6561_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6562_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6563_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6564_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6565_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6566_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6567_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6568_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),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),set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6569_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),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),set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6570_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6571_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,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))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6572_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,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),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6573_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,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),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6574_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6575_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6576_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6577_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6578_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_6579_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_6580_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_6581_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_6582_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_6583_finite__remove__induct,axiom,
    ! [B2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [A5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A5))
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A5),B2))
                 => ( ! [X: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A5))
                       => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) )
                   => pp(aa_set8260762808118238015T_bool(P,A5)) ) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6584_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
     => ( pp(aa_set_set_nat_bool(P,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),B2))
                 => ( ! [X: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X),A5))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A5)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6585_finite__remove__induct,axiom,
    ! [B2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(P,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),B2))
                 => ( ! [X: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X),A5))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A5,insert_complex(X,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A5)) ) ) ) )
         => pp(aa_set_complex_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6586_finite__remove__induct,axiom,
    ! [B2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(P,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),B2))
                 => ( ! [X: real] :
                        ( pp(aa_set_real_bool(member_real(X),A5))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A5,insert_real(X,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A5)) ) ) ) )
         => pp(aa_set_real_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6587_finite__remove__induct,axiom,
    ! [B2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(P,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),B2))
                 => ( ! [X: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X),A5))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A5,insert_nat(X,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A5)) ) ) ) )
         => pp(aa_set_nat_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6588_finite__remove__induct,axiom,
    ! [B2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(P,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),B2))
                 => ( ! [X: int] :
                        ( pp(aa_set_int_bool(member_int(X),A5))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A5,insert_int(X,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A5)) ) ) ) )
         => pp(aa_set_int_bool(P,B2)) ) ) ) ).

% finite_remove_induct
tff(fact_6589_remove__induct,axiom,
    ! [P: fun_se448528567104302406T_bool,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
     => ( ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
         => pp(aa_set8260762808118238015T_bool(P,B2)) )
       => ( ! [A5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A5))
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A5),B2))
                 => ( ! [X: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A5))
                       => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) )
                   => pp(aa_set8260762808118238015T_bool(P,A5)) ) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6590_remove__induct,axiom,
    ! [P: fun_set_set_nat_bool,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
     => ( ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B2))
         => pp(aa_set_set_nat_bool(P,B2)) )
       => ( ! [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),B2))
                 => ( ! [X: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X),A5))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A5,insert_set_nat(X,bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A5)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6591_remove__induct,axiom,
    ! [P: fun_set_complex_bool,B2: set_complex] :
      ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
     => ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
         => pp(aa_set_complex_bool(P,B2)) )
       => ( ! [A5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
             => ( ( A5 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A5),B2))
                 => ( ! [X: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X),A5))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A5,insert_complex(X,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A5)) ) ) ) )
         => pp(aa_set_complex_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6592_remove__induct,axiom,
    ! [P: fun_set_real_bool,B2: set_real] :
      ( pp(aa_set_real_bool(P,bot_bot_set_real))
     => ( ( ~ pp(aa_set_real_bool(finite_finite_real,B2))
         => pp(aa_set_real_bool(P,B2)) )
       => ( ! [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),B2))
                 => ( ! [X: real] :
                        ( pp(aa_set_real_bool(member_real(X),A5))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A5,insert_real(X,bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A5)) ) ) ) )
         => pp(aa_set_real_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6593_remove__induct,axiom,
    ! [P: fun_set_nat_bool,B2: set_nat] :
      ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
     => ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B2))
         => pp(aa_set_nat_bool(P,B2)) )
       => ( ! [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),B2))
                 => ( ! [X: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X),A5))
                       => pp(aa_set_nat_bool(P,minus_minus_set_nat(A5,insert_nat(X,bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A5)) ) ) ) )
         => pp(aa_set_nat_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6594_remove__induct,axiom,
    ! [P: fun_set_int_bool,B2: set_int] :
      ( pp(aa_set_int_bool(P,bot_bot_set_int))
     => ( ( ~ pp(aa_set_int_bool(finite_finite_int,B2))
         => pp(aa_set_int_bool(P,B2)) )
       => ( ! [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),B2))
                 => ( ! [X: int] :
                        ( pp(aa_set_int_bool(member_int(X),A5))
                       => pp(aa_set_int_bool(P,minus_minus_set_int(A5,insert_int(X,bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A5)) ) ) ) )
         => pp(aa_set_int_bool(P,B2)) ) ) ) ).

% remove_induct
tff(fact_6595_finite__induct__select,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [T5: set_complex] :
              ( pp(aa_set_complex_bool(ord_less_set_complex(T5),S2))
             => ( pp(aa_set_complex_bool(P,T5))
               => ? [X: complex] :
                    ( pp(aa_set_complex_bool(member_complex(X),minus_811609699411566653omplex(S2,T5)))
                    & pp(aa_set_complex_bool(P,insert_complex(X,T5))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6596_finite__induct__select,axiom,
    ! [S2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [T5: set_real] :
              ( pp(aa_set_real_bool(ord_less_set_real(T5),S2))
             => ( pp(aa_set_real_bool(P,T5))
               => ? [X: real] :
                    ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(S2,T5)))
                    & pp(aa_set_real_bool(P,insert_real(X,T5))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6597_finite__induct__select,axiom,
    ! [S2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [T5: set_int] :
              ( pp(aa_set_int_bool(ord_less_set_int(T5),S2))
             => ( pp(aa_set_int_bool(P,T5))
               => ? [X: int] :
                    ( pp(aa_set_int_bool(member_int(X),minus_minus_set_int(S2,T5)))
                    & pp(aa_set_int_bool(P,insert_int(X,T5))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6598_finite__induct__select,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [T5: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,T5),S2))
             => ( pp(aa_set_nat_bool(P,T5))
               => ? [X: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X),minus_minus_set_nat(S2,T5)))
                    & pp(aa_set_nat_bool(P,insert_nat(X,T5))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_6599_psubset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,B2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),insert_VEBT_VEBT(X2,B2)))
    <=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
         => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),B2)) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2))
         => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT))),B2)) )
            & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
             => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6600_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),insert_set_nat(X2,B2)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X2),B2))
         => pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B2)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),B2))
         => ( ( pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
             => pp(aa_set_set_nat_bool(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X2,bot_bot_set_set_nat))),B2)) )
            & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X2),A2))
             => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6601_psubset__insert__iff,axiom,
    ! [A2: set_real,X2: real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),insert_real(X2,B2)))
    <=> ( ( pp(aa_set_real_bool(member_real(X2),B2))
         => pp(aa_set_real_bool(ord_less_set_real(A2),B2)) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),B2))
         => ( ( pp(aa_set_real_bool(member_real(X2),A2))
             => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real))),B2)) )
            & ( ~ pp(aa_set_real_bool(member_real(X2),A2))
             => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6602_psubset__insert__iff,axiom,
    ! [A2: set_nat,X2: nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),insert_nat(X2,B2)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X2),B2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B2)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),B2))
         => ( ( pp(aa_set_nat_bool(member_nat(X2),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,minus_minus_set_nat(A2,insert_nat(X2,bot_bot_set_nat))),B2)) )
            & ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6603_psubset__insert__iff,axiom,
    ! [A2: set_int,X2: int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),insert_int(X2,B2)))
    <=> ( ( pp(aa_set_int_bool(member_int(X2),B2))
         => pp(aa_set_int_bool(ord_less_set_int(A2),B2)) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),B2))
         => ( ( pp(aa_set_int_bool(member_int(X2),A2))
             => pp(aa_set_int_bool(ord_less_set_int(minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int))),B2)) )
            & ( ~ pp(aa_set_int_bool(member_int(X2),A2))
             => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_6604_set__replicate__Suc,axiom,
    ! [N: nat,X2: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X2)) = insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_6605_set__replicate__Suc,axiom,
    ! [N: nat,X2: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X2)) = insert_real(X2,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_6606_set__replicate__Suc,axiom,
    ! [N: nat,X2: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X2)) = insert_nat(X2,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_6607_set__replicate__Suc,axiom,
    ! [N: nat,X2: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X2)) = insert_int(X2,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_6608_set__replicate__conv__if,axiom,
    ! [N: nat,X2: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X2)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X2)) = insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_6609_set__replicate__conv__if,axiom,
    ! [N: nat,X2: real] :
      ( ( ( N = zero_zero_nat )
       => ( set_real2(replicate_real(N,X2)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( set_real2(replicate_real(N,X2)) = insert_real(X2,bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_6610_set__replicate__conv__if,axiom,
    ! [N: nat,X2: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X2)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X2)) = insert_nat(X2,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_6611_set__replicate__conv__if,axiom,
    ! [N: nat,X2: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X2)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X2)) = insert_int(X2,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_6612_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_lf(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6613_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,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,M)),groups3539618377306564664at_int(aTP_Lamm_lg(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6614_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,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,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kl(G)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6615_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_km(G)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6616_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_lh(F),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6617_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(aTP_Lamm_li(F),set_or1269000886237332187st_nat(M,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,M)) ) ) ).

% sum_Suc_diff
tff(fact_6618_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lj(F)),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6619_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6620_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6621_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => 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,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6622_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6623_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6624_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => 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,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6625_sum__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2)))
             => 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,A2)),groups2906978787729119204at_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6626_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6627_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6628_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => 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,A2)),groups5693394587270226106ex_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6629_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_6630_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_6631_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,N))),groups2906978787729119204at_rat(G,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),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6632_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,N))),groups3539618377306564664at_int(G,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),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6633_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),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),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6634_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,N))),aa_set_nat_real(groups6591440286371151544t_real(G),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),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6635_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6636_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),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,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6637_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),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,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6638_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6639_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_6640_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_6641_Suc__div__eq__add3__div,axiom,
    ! [M: 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,M)))),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))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_6642_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_6643_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),X6))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,X6))
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X6) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_6644_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_6645_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_6646_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_6647_Suc__mod__eq__add3__mod,axiom,
    ! [M: 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,M)))),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))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_6648_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),minus_5127226145743854075T_VEBT(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,B)))
           => ( ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6649_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( 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),B2))
                 => 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,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6650_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( 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),B2))
                 => 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,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6651_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),minus_5127226145743854075T_VEBT(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B)))
           => ( ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6652_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( 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),B2))
                 => 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,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6653_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( 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),B2))
                 => 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,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6654_sum__strict__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,B: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( pp(aa_set_nat_bool(member_nat(B),minus_minus_set_nat(B2,A2)))
         => ( 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),B2))
                 => 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,A2)),groups2906978787729119204at_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6655_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B),minus_5127226145743854075T_VEBT(B2,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B)))
           => ( ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),B2))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6656_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( 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),B2))
                 => 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,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6657_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( 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),B2))
                 => 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,A2)),groups5693394587270226106ex_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6658_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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,M),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,M),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,M),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_6659_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_lk(F),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(F,M)),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,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_lk(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_6660_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_ll(F),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),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,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_ll(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_6661_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_lm(F),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),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,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_lm(F),set_or1269000886237332187st_nat(M,N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_6662_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ln(F)),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),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,M),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ln(F)),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_6663_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_lo(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_6664_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_lp(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_6665_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lq(F)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_6666_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_6667_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) = 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_ay,N))) ).

% mask_eq_sum_exp
tff(fact_6668_mask__eq__sum__exp,axiom,
    ! [N: nat] : minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer) = groups7501900531339628137nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,N))) ).

% mask_eq_sum_exp
tff(fact_6669_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_ay,N))) ).

% mask_eq_sum_exp
tff(fact_6670_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X2)),groups2073611262835488442omplex(power_power_complex(X2),set_or1269000886237332187st_nat(M,N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X2),M)),aa_nat_complex(power_power_complex(X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6671_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),M)),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6672_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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),X2)),groups3539618377306564664at_int(power_power_int(X2),set_or1269000886237332187st_nat(M,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X2),M)),aa_nat_int(power_power_int(X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6673_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),set_or1269000886237332187st_nat(M,N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X2),M)),aa_nat_real(power_power_real(X2),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_6674_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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_lr(G),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_6675_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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)))) = groups3539618377306564664at_int(aTP_Lamm_ls(G),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_6676_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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_lt(G)),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_6677_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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_lu(G)),set_or1269000886237332187st_nat(M,N)) ).

% sum.in_pairs
tff(fact_6678_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),numera6620942414471956472nteger(aa_num_num(bit1,K))) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(N,numera6620942414471956472nteger(K)),numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).

% take_bit_Suc_bit1
tff(fact_6679_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: 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,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_6680_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: 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,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_6681_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_ay,N))) ).

% mask_eq_sum_exp_nat
tff(fact_6682_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),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_6683_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lv(A),D)),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_6684_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6685_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6686_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6687_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6688_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6689_sum__gp,axiom,
    ! [N: nat,M: nat,X2: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(power_power_complex(X2),set_or1269000886237332187st_nat(M,N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X2 = one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X2),set_or1269000886237332187st_nat(M,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)),M)) ) )
          & ( ( X2 != one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X2),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X2),M)),aa_nat_complex(power_power_complex(X2),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X2)) ) ) ) ) ) ).

% sum_gp
tff(fact_6690_sum__gp,axiom,
    ! [N: nat,M: nat,X2: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X2 = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,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)),M)) ) )
          & ( ( X2 != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),M)),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ) ) ).

% sum_gp
tff(fact_6691_sum__gp,axiom,
    ! [N: nat,M: nat,X2: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),set_or1269000886237332187st_nat(M,N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X2 = one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),set_or1269000886237332187st_nat(M,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)),M)) ) )
          & ( ( X2 != one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X2),M)),aa_nat_real(power_power_real(X2),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ) ) ).

% sum_gp
tff(fact_6692_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6693_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6694_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6695_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6696_dbl__inc__simps_I3_J,axiom,
    neg_nu5831290666863070958nteger(one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6697_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = divide6298287555418463151nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6698_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,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_6699_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_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_6700_sum__gp__offset,axiom,
    ! [X2: complex,M: nat,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X2 != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X2),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X2),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X2)) ) ) ) ).

% sum_gp_offset
tff(fact_6701_sum__gp__offset,axiom,
    ! [X2: rat,M: nat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat) ) )
      & ( ( X2 != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),M)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N))))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ).

% sum_gp_offset
tff(fact_6702_sum__gp__offset,axiom,
    ! [X2: real,M: nat,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X2 != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X2),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X2),aa_nat_nat(suc,N))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ).

% sum_gp_offset
tff(fact_6703_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),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_6704_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,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_6705_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),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_6706_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N))) = times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6707_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))),groups3539618377306564664at_int(semiri1314217659103216013at_int,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_6708_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),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_6709_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),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_6710_gauss__sum,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,N)) = divide6298287555418463151nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6711_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,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_6712_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_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_6713_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_6714_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_6715_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_6716_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_6717_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_6718_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_6719_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_6720_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_6721_negative__zle,axiom,
    ! [N: nat,M: 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,M))) ).

% negative_zle
tff(fact_6722_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% int_dvd_int_iff
tff(fact_6723_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_6724_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_6725_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_6726_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_6727_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_6728_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_6729_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_6730_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_6731_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_6732_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_6733_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_6734_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_6735_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_6736_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_6737_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6738_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6739_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6740_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6741_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6742_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_6743_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_6744_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_6745_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_6746_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_6747_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_6748_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_6749_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_6750_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_num_nat(numeral_numeral_nat,N)) = numera1916890842035813515d_enat(N) ).

% of_nat_numeral
tff(fact_6751_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,N)) = numera6620942414471956472nteger(N) ).

% of_nat_numeral
tff(fact_6752_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% of_nat_numeral
tff(fact_6753_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = aa_num_real(numeral_numeral_real,N) ).

% of_nat_numeral
tff(fact_6754_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).

% of_nat_numeral
tff(fact_6755_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_6756_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_6757_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_6758_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_6759_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_6760_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_6761_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_6762_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_6763_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_6764_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6765_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6766_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6767_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6768_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_6769_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_6770_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_6771_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_6772_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_6773_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_6774_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_6775_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_6776_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_6777_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_6778_negative__zless,axiom,
    ! [N: nat,M: 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,M))) ).

% negative_zless
tff(fact_6779_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_6780_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_6781_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_6782_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_6783_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat(P)) = zero_n3304061248610475627l_real(P) ).

% of_nat_of_bool
tff(fact_6784_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat(P)) = zero_n2687167440665602831ol_nat(P) ).

% of_nat_of_bool
tff(fact_6785_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat(P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% of_nat_of_bool
tff(fact_6786_dbl__inc__simps_I4_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_inc_simps(4)
tff(fact_6787_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_6788_dbl__inc__simps_I4_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_inc_simps(4)
tff(fact_6789_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_6790_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_6791_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6792_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6793_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K)) = numera6620942414471956472nteger(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6794_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_lw(F),A2) ).

% of_nat_sum
tff(fact_6795_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_lx(F),A2) ).

% of_nat_sum
tff(fact_6796_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_ly(F),A2) ).

% of_nat_sum
tff(fact_6797_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lz(F)),A2) ).

% of_nat_sum
tff(fact_6798_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ma(F)),A2) ).

% of_nat_sum
tff(fact_6799_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6800_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6801_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6802_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6803_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_6804_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_6805_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_6806_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_6807_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_6808_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_6809_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_6810_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_6811_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_6812_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_6813_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_6814_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_6815_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6816_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_6817_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_6818_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_6819_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K))) ).

% dbl_dec_simps(1)
tff(fact_6820_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_6821_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_6822_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_6823_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K))) ).

% dbl_inc_simps(1)
tff(fact_6824_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_6825_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_6826_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_6827_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_6828_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6829_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6830_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6831_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6832_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: 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,X2)))
    <=> 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)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6833_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: 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,X2)))
    <=> 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)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6834_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> 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)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6835_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: 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,X2)))
    <=> 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)),X2)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6836_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> 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)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6837_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: 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,X2)))
    <=> 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)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6838_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: 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,X2)))
    <=> 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)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6839_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X2: 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,X2)))
    <=> 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)),X2)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6840_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_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,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6841_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6842_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6843_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),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,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6844_of__nat__zero__less__power__iff,axiom,
    ! [X2: 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,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6845_of__nat__zero__less__power__iff,axiom,
    ! [X2: 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,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6846_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6847_of__nat__zero__less__power__iff,axiom,
    ! [X2: 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,X2)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6848_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6849_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6850_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6851_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6852_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6853_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: 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,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6854_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6855_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: 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,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6856_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6857_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: 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,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6858_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6859_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6860_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6861_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6862_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6863_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6864_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N)),aa_nat_real(semiri5074537144036343181t_real,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6865_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: 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,I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6866_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: 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,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6867_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: 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,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X2)))
    <=> 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,I)),N)),X2)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6868_real__arch__simple,axiom,
    ! [X2: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% real_arch_simple
tff(fact_6869_real__arch__simple,axiom,
    ! [X2: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% real_arch_simple
tff(fact_6870_reals__Archimedean2,axiom,
    ! [X2: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% reals_Archimedean2
tff(fact_6871_reals__Archimedean2,axiom,
    ! [X2: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% reals_Archimedean2
tff(fact_6872_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X2)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X2)) ).

% mult_of_nat_commute
tff(fact_6873_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X2)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X2)) ).

% mult_of_nat_commute
tff(fact_6874_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X2)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X2)) ).

% mult_of_nat_commute
tff(fact_6875_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X2)) ).

% mult_of_nat_commute
tff(fact_6876_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% int_cases2
tff(fact_6877_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_6878_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: 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,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X2)) ) ).

% of_nat_less_of_int_iff
tff(fact_6879_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: 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,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X2)) ) ).

% of_nat_less_of_int_iff
tff(fact_6880_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: 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,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X2)) ) ).

% of_nat_less_of_int_iff
tff(fact_6881_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_6882_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_6883_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_6884_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_6885_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_6886_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_6887_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_6888_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_6889_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_6890_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_6891_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_6892_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_6893_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_6894_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_6895_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_6896_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_6897_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_6898_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_6899_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_6900_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_6901_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_6902_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_6903_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_6904_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_6905_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_6906_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_6907_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_6908_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_6909_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z: int] :
      ( ! [N2: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))))
       => pp(aa_int_bool(P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_6910_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_6911_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_6912_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% zle_int
tff(fact_6913_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_6914_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_6915_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_6916_int__plus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_6917_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),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,M),N))),Z) ).

% zadd_int_left
tff(fact_6918_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_6919_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_6920_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z))
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% zle_iff_zadd
tff(fact_6921_not__int__zless__negative,axiom,
    ! [N: nat,M: 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,M)))) ).

% not_int_zless_negative
tff(fact_6922_of__nat__max,axiom,
    ! [X2: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_6923_of__nat__max,axiom,
    ! [X2: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_6924_of__nat__max,axiom,
    ! [X2: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X2)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_6925_nat__less__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_6926_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_6927_ex__less__of__nat__mult,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => ? [N2: 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,N2)),X2))) ) ).

% ex_less_of_nat_mult
tff(fact_6928_ex__less__of__nat__mult,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ? [N2: 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,N2)),X2))) ) ).

% ex_less_of_nat_mult
tff(fact_6929_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_6930_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_6931_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_6932_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_6933_reals__Archimedean3,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ! [Y3: real] :
        ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X2))) ) ).

% reals_Archimedean3
tff(fact_6934_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_6935_int__zle__neg,axiom,
    ! [N: nat,M: 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,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_6936_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_6937_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_6938_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_6939_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_6940_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_6941_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_lw(F),A2) ).

% int_sum
tff(fact_6942_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_ly(F),A2) ).

% int_sum
tff(fact_6943_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_cd,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_cu(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_6944_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_cd,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_6945_mod__mult2__eq_H,axiom,
    ! [A: int,M: 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,M)),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,M)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_6946_mod__mult2__eq_H,axiom,
    ! [A: nat,M: 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,M)),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,M)),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,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_6947_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_6948_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% pos_int_cases
tff(fact_6949_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).

% int_cases3
tff(fact_6950_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> 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,M))) ) ).

% nat_less_real_le
tff(fact_6951_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
    <=> 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,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_6952_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => 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,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_6953_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_6954_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_6955_negD,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
     => ? [N2: nat] : X2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_6956_dbl__inc__def,axiom,
    ! [X2: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X2),X2)),one_one_complex) ).

% dbl_inc_def
tff(fact_6957_dbl__inc__def,axiom,
    ! [X2: real] : aa_real_real(neg_nu8295874005876285629c_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),X2)),one_one_real) ).

% dbl_inc_def
tff(fact_6958_dbl__inc__def,axiom,
    ! [X2: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(X2),X2)),one_one_rat) ).

% dbl_inc_def
tff(fact_6959_dbl__inc__def,axiom,
    ! [X2: int] : aa_int_int(neg_nu5851722552734809277nc_int,X2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),X2)),one_one_int) ).

% dbl_inc_def
tff(fact_6960_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_6961_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,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),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_6962_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_6963_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
     => ~ ! [N2: 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,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_6964_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [N2: 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,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_6965_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_6966_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6967_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_6968_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(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_6969_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( 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,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_6970_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( 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,M))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_6971_real__archimedian__rdiv__eq__0,axiom,
    ! [X2: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M3))
             => 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,M3)),X2)),C)) )
         => ( X2 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_6972_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% neg_int_cases
tff(fact_6973_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X2: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_6974_ln__realpow,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(ln_ln_real,aa_nat_real(power_power_real(X2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X2)) ) ) ).

% ln_realpow
tff(fact_6975_dbl__dec__def,axiom,
    ! [X2: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X2) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X2),X2)),one_one_complex) ).

% dbl_dec_def
tff(fact_6976_dbl__dec__def,axiom,
    ! [X2: real] : aa_real_real(neg_nu6075765906172075777c_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),X2)),one_one_real) ).

% dbl_dec_def
tff(fact_6977_dbl__dec__def,axiom,
    ! [X2: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(X2),X2)),one_one_rat) ).

% dbl_dec_def
tff(fact_6978_dbl__dec__def,axiom,
    ! [X2: int] : aa_int_int(neg_nu3811975205180677377ec_int,X2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),X2)),one_one_int) ).

% dbl_dec_def
tff(fact_6979_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,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_6980_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,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_6981_double__gauss__sum,axiom,
    ! [N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,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_6982_double__gauss__sum,axiom,
    ! [N: nat] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,N))) = times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_6983_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))),groups3539618377306564664at_int(semiri1314217659103216013at_int,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_6984_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),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_6985_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),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_6986_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_mb(A),D),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_6987_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_mc(A),D),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_6988_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_md(A),D),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_6989_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_me(A),D),set_or1269000886237332187st_nat(zero_zero_nat,N))) = times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A),times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),D))) ).

% double_arith_series
tff(fact_6990_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))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_mf(A),D),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_6991_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_mg(A),D)),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_6992_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_mh(A),D)),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_6993_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_me(A),D),set_or1269000886237332187st_nat(zero_zero_nat,N)) = divide6298287555418463151nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),A),times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),D))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6994_arith__series,axiom,
    ! [A: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_mf(A),D),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_6995_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mg(A),D)),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_6996_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_mi,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6997_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_mj,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6998_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_mk,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_6999_of__nat__code__if,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = zero_z3403309356797280102nteger ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = produc1830744345554046123nteger(aTP_Lamm_ml,divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_7000_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_mm),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_7001_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_mn),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_7002_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_mo),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% of_nat_code_if
tff(fact_7003_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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_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(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(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_7004_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(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_7005_ln__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X2) = suminf_real(aTP_Lamm_mp(X2)) ) ) ) ).

% ln_series
tff(fact_7006_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_mr(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_7007_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_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_7008_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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_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(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_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_7009_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_7010_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_7011_pochhammer__double,axiom,
    ! [Z: real,N: nat] : aa_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(comm_s7457072308508201937r_real(Z),N))),aa_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_7012_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_mw,N),zero_zero_complex) ).

% of_nat_code
tff(fact_7013_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_mx,N),zero_zero_rat) ).

% of_nat_code
tff(fact_7014_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_my),N),zero_zero_int) ).

% of_nat_code
tff(fact_7015_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_mz),N),zero_zero_real) ).

% of_nat_code
tff(fact_7016_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_na),N),zero_zero_nat) ).

% of_nat_code
tff(fact_7017_lessThan__eq__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,X2) = aa_nat_set_nat(set_ord_lessThan_nat,Y) )
    <=> ( X2 = Y ) ) ).

% lessThan_eq_iff
tff(fact_7018_lessThan__eq__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( set_ord_lessThan_int(X2) = set_ord_lessThan_int(Y) )
    <=> ( X2 = Y ) ) ).

% lessThan_eq_iff
tff(fact_7019_lessThan__eq__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( set_or5984915006950818249n_real(X2) = set_or5984915006950818249n_real(Y) )
    <=> ( X2 = Y ) ) ).

% lessThan_eq_iff
tff(fact_7020_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or890127255671739683et_nat(K)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,I),K)) ) ).

% lessThan_iff
tff(fact_7021_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_ord_lessThan_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),K)) ) ).

% lessThan_iff
tff(fact_7022_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( pp(aa_set_num_bool(member_num(I),set_ord_lessThan_num(K)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),K)) ) ).

% lessThan_iff
tff(fact_7023_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_7024_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_lessThan_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) ) ).

% lessThan_iff
tff(fact_7025_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or5984915006950818249n_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).

% lessThan_iff
tff(fact_7026_finite__lessThan,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K))) ).

% finite_lessThan
tff(fact_7027_lessThan__subset__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_lessThan_rat(X2)),set_ord_lessThan_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ).

% lessThan_subset_iff
tff(fact_7028_lessThan__subset__iff,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_lessThan_num(X2)),set_ord_lessThan_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y)) ) ).

% lessThan_subset_iff
tff(fact_7029_lessThan__subset__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,X2)),aa_nat_set_nat(set_ord_lessThan_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ).

% lessThan_subset_iff
tff(fact_7030_lessThan__subset__iff,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X2)),set_ord_lessThan_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y)) ) ).

% lessThan_subset_iff
tff(fact_7031_lessThan__subset__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X2)),set_or5984915006950818249n_real(Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ).

% lessThan_subset_iff
tff(fact_7032_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_7033_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(comm_s7457072308508201937r_real(A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_7034_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_7035_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_7036_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_7037_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_7038_single__Diff__lessThan,axiom,
    ! [K: nat] : minus_minus_set_nat(insert_nat(K,bot_bot_set_nat),aa_nat_set_nat(set_ord_lessThan_nat,K)) = insert_nat(K,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_7039_single__Diff__lessThan,axiom,
    ! [K: int] : minus_minus_set_int(insert_int(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_7040_single__Diff__lessThan,axiom,
    ! [K: real] : minus_minus_set_real(insert_real(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real(K,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_7041_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_7042_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : 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,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% sum.lessThan_Suc
tff(fact_7043_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_7044_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_7045_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_gn(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7046_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_gp(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7047_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_7048_int__if,axiom,
    ! [P: bool,A: nat,B: nat] :
      ( ( pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,A) ) )
      & ( ~ pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ) ).

% int_if
tff(fact_7049_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% int_int_eq
tff(fact_7050_lessThan__non__empty,axiom,
    ! [X2: int] : set_ord_lessThan_int(X2) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_7051_lessThan__non__empty,axiom,
    ! [X2: real] : set_or5984915006950818249n_real(X2) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_7052_infinite__Iio,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A))) ).

% infinite_Iio
tff(fact_7053_infinite__Iio,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(finite_finite_real,set_or5984915006950818249n_real(A))) ).

% infinite_Iio
tff(fact_7054_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_nb,U)) ).

% lessThan_def
tff(fact_7055_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_nc,U)) ).

% lessThan_def
tff(fact_7056_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_nd,U)) ).

% lessThan_def
tff(fact_7057_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,U)) ).

% lessThan_def
tff(fact_7058_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_ne,U)) ).

% lessThan_def
tff(fact_7059_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_nf,U)) ).

% lessThan_def
tff(fact_7060_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_7061_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(N)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7062_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7063_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7064_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7065_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(N)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_7066_pochhammer__pos,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X2),N))) ) ).

% pochhammer_pos
tff(fact_7067_pochhammer__pos,axiom,
    ! [X2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X2),N))) ) ).

% pochhammer_pos
tff(fact_7068_pochhammer__pos,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
     => 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,X2),N))) ) ).

% pochhammer_pos
tff(fact_7069_pochhammer__pos,axiom,
    ! [X2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X2),N))) ) ).

% pochhammer_pos
tff(fact_7070_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M: 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),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7071_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7072_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: 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),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7073_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7074_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),M) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7075_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7076_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_7077_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_7078_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ).

% finite_nat_iff_bounded
tff(fact_7079_finite__nat__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_bounded
tff(fact_7080_pochhammer__nonneg,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7081_pochhammer__nonneg,axiom,
    ! [X2: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7082_pochhammer__nonneg,axiom,
    ! [X2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X2))
     => 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,X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7083_pochhammer__nonneg,axiom,
    ! [X2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X2),N))) ) ).

% pochhammer_nonneg
tff(fact_7084_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_7085_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_7086_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_7087_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_7088_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_7089_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ng(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_7090_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nh(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_7091_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: 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(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,set_ord_lessThan_int(N))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(N))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kh(Q),P),set_ord_lessThan_int(N)) ) ) ).

% sum_diff_distrib
tff(fact_7092_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: 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(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(N))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(N))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kj(Q),P),set_or5984915006950818249n_real(N)) ) ) ).

% sum_diff_distrib
tff(fact_7093_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: 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(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,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_kk(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_7094_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_7095_pochhammer__rec,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_7096_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_7097_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_7098_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_7099_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_7100_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_7101_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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_7102_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_7103_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_7104_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_7105_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] : aa_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(comm_s7457072308508201937r_real(Z),N)) ).

% pochhammer_rec'
tff(fact_7106_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_7107_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7108_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7109_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
    <=> ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7110_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7111_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7112_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7113_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7114_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7115_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7116_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7117_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7118_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7119_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7120_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7121_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7122_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7123_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7124_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7125_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = 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))),M)) ).

% pochhammer_product'
tff(fact_7126_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = 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))),M)) ).

% pochhammer_product'
tff(fact_7127_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),N)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).

% pochhammer_product'
tff(fact_7128_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: 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),M)) = 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))),M)) ).

% pochhammer_product'
tff(fact_7129_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_lf(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7130_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : 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)),groups3539618377306564664at_int(aTP_Lamm_lg(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7131_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_kl(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7132_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_km(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_7133_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_ni(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7134_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_nj(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7135_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nk(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7136_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_lh(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7137_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_li(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7138_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lj(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7139_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kl(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_7140_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_km(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_7141_one__diff__power__eq,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X2),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X2)),groups2073611262835488442omplex(power_power_complex(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7142_one__diff__power__eq,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7143_one__diff__power__eq,axiom,
    ! [X2: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X2),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),X2)),groups3539618377306564664at_int(power_power_int(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7144_one__diff__power__eq,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_7145_power__diff__1__eq,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X2),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X2),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7146_power__diff__1__eq,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),N)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7147_power__diff__1__eq,axiom,
    ! [X2: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X2),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,X2),one_one_int)),groups3539618377306564664at_int(power_power_int(X2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7148_power__diff__1__eq,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X2),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_7149_geometric__sum,axiom,
    ! [X2: complex,N: nat] :
      ( ( X2 != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(X2),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(X2),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X2),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_7150_geometric__sum,axiom,
    ! [X2: rat,N: nat] :
      ( ( X2 != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X2),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(X2),N)),one_one_rat)),aa_rat_rat(minus_minus_rat(X2),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_7151_geometric__sum,axiom,
    ! [X2: real,N: nat] :
      ( ( X2 != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),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(power_power_real(X2),N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),one_one_real)) ) ) ).

% geometric_sum
tff(fact_7152_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7153_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7154_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),M)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7155_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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),M)),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,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_7156_sum__gp__strict,axiom,
    ! [X2: complex,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X2 != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X2),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(X2),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X2)) ) ) ) ).

% sum_gp_strict
tff(fact_7157_sum__gp__strict,axiom,
    ! [X2: rat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
      & ( ( X2 != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),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(X2),N))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ).

% sum_gp_strict
tff(fact_7158_sum__gp__strict,axiom,
    ! [X2: real,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X2 != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),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(power_power_real(X2),N))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ).

% sum_gp_strict
tff(fact_7159_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nl(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nm(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7160_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nn(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_no(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7161_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_np(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nq(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7162_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nr(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ns(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7163_power__diff__sumr2,axiom,
    ! [X2: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X2),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X2),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nt(X2),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7164_power__diff__sumr2,axiom,
    ! [X2: rat,N: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),N)),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nu(X2),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7165_power__diff__sumr2,axiom,
    ! [X2: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X2),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,X2),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nv(X2),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7166_power__diff__sumr2,axiom,
    ! [X2: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X2),N)),aa_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,X2),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nw(X2),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_7167_diff__power__eq__sum,axiom,
    ! [X2: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X2),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(X2),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nx(X2),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7168_diff__power__eq__sum,axiom,
    ! [X2: rat,N: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X2),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(X2),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ny(X2),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7169_diff__power__eq__sum,axiom,
    ! [X2: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X2),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,X2),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nz(X2),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7170_diff__power__eq__sum,axiom,
    ! [X2: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X2),aa_nat_nat(suc,N))),aa_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,X2),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_oa(X2),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_7171_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_7172_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_7173_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : times_3573771949741848930nteger(minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = times_3573771949741848930nteger(R2,aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_7174_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_7175_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_7176_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K5: rat,K: 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),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7177_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K5: int,K: 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,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),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_7178_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K5: nat,K: 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),K)))),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_7179_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K5: real,K: 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),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7180_one__diff__power__eq_H,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X2),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X2)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ob(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7181_one__diff__power__eq_H,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oc(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7182_one__diff__power__eq_H,axiom,
    ! [X2: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X2),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),X2)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_od(X2),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7183_one__diff__power__eq_H,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oe(X2),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_7184_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_7185_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_7186_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_7187_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_7188_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_7189_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_7190_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_7191_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_7192_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_7193_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_7194_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_of(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_og(F)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_7195_norm__le__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X2)),zero_zero_real))
    <=> ( X2 = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_7196_norm__le__zero__iff,axiom,
    ! [X2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X2)),zero_zero_real))
    <=> ( X2 = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_7197_zero__less__norm__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X2)))
    <=> ( X2 != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_7198_zero__less__norm__iff,axiom,
    ! [X2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X2)))
    <=> ( X2 != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_7199_suminf__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => ( suminf_real(power_power_real(C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7200_suminf__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_7201_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_7202_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_7203_norm__eq__zero,axiom,
    ! [X2: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X2) = zero_zero_real )
    <=> ( X2 = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_7204_norm__eq__zero,axiom,
    ! [X2: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X2) = zero_zero_real )
    <=> ( X2 = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_7205_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_oi) = zero_zero_complex ).

% suminf_zero
tff(fact_7206_suminf__zero,axiom,
    suminf_real(aTP_Lamm_fi) = zero_zero_real ).

% suminf_zero
tff(fact_7207_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_cb) = zero_zero_nat ).

% suminf_zero
tff(fact_7208_suminf__zero,axiom,
    suminf_int(aTP_Lamm_oj) = zero_zero_int ).

% suminf_zero
tff(fact_7209_norm__not__less__zero,axiom,
    ! [X2: complex] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),zero_zero_real)) ).

% norm_not_less_zero
tff(fact_7210_norm__uminus__minus,axiom,
    ! [X2: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X2)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),Y)) ).

% norm_uminus_minus
tff(fact_7211_norm__uminus__minus,axiom,
    ! [X2: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X2)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X2),Y)) ).

% norm_uminus_minus
tff(fact_7212_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_7213_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_7214_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z: real] :
      ( ( aa_nat_real(power_power_real(W2),N) = aa_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_7215_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_7216_norm__mult__less,axiom,
    ! [X2: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),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,X2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_7217_norm__mult__less,axiom,
    ! [X2: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),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(X2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_7218_norm__triangle__lt,axiom,
    ! [X2: 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,X2)),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,X2),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_7219_norm__triangle__lt,axiom,
    ! [X2: 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,X2)),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(X2),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_7220_norm__add__less,axiom,
    ! [X2: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),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,X2),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_7221_norm__add__less,axiom,
    ! [X2: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),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(X2),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_7222_norm__add__leD,axiom,
    ! [A: real,B: real,C: 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))),C))
     => 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)),C))) ) ).

% norm_add_leD
tff(fact_7223_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: 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))),C))
     => 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)),C))) ) ).

% norm_add_leD
tff(fact_7224_norm__triangle__le,axiom,
    ! [X2: 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,X2)),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,X2),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_7225_norm__triangle__le,axiom,
    ! [X2: 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,X2)),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(X2),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_7226_norm__triangle__ineq,axiom,
    ! [X2: 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,X2),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X2)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_7227_norm__triangle__ineq,axiom,
    ! [X2: 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(X2),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X2)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_7228_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_7229_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_7230_norm__diff__triangle__less,axiom,
    ! [X2: 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,X2),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,X2),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_7231_norm__diff__triangle__less,axiom,
    ! [X2: 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(X2),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(X2),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_7232_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_7233_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_7234_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N4) ) ) ) ).

% suminf_finite
tff(fact_7235_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N4) ) ) ) ).

% suminf_finite
tff(fact_7236_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N4) ) ) ) ).

% suminf_finite
tff(fact_7237_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N4) ) ) ) ).

% suminf_finite
tff(fact_7238_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( aa_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_7239_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_7240_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: 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,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_7241_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: 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(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_7242_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_ok(Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7243_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_ol(Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7244_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,N))),aa_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(aTP_Lamm_om(Z)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7245_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_on) ).

% pi_series
tff(fact_7246_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( 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(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(K),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))),K))),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(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(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(K),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))),K))),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(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_7247_and__int_Oelims,axiom,
    ! [X2: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X2),Xa2) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X2),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))),X2))),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(X2),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))),X2))),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(X2),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_7248_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(set_fo1517530859248394432omplex(aTP_Lamm_oo(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_7249_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_op(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_7250_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_oq(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_7251_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_or(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_7252_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_os(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_7253_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(aTP_Lamm_ot(Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_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_7254_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_ou(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_7255_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_7256_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_7257_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_7258_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_7259_bit_Oconj__zero__left,axiom,
    ! [X2: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X2) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_7260_bit_Oconj__zero__right,axiom,
    ! [X2: int] : aa_int_int(bit_se725231765392027082nd_int(X2),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_7261_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_lw(F),A2) ).

% of_nat_prod
tff(fact_7262_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ma(F)),A2) ).

% of_nat_prod
tff(fact_7263_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_ly(F),A2) ).

% of_nat_prod
tff(fact_7264_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_lz(F)),A2) ).

% of_nat_prod
tff(fact_7265_sums__zero,axiom,
    pp(aa_complex_bool(sums_complex(aTP_Lamm_oi),zero_zero_complex)) ).

% sums_zero
tff(fact_7266_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_fi),zero_zero_real)) ).

% sums_zero
tff(fact_7267_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_cb),zero_zero_nat)) ).

% sums_zero
tff(fact_7268_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_oj),zero_zero_int)) ).

% sums_zero
tff(fact_7269_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gj(F)),A2) ).

% of_int_prod
tff(fact_7270_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A2)) = groups73079841787564623at_rat(aTP_Lamm_ov(F),A2) ).

% of_int_prod
tff(fact_7271_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A2)) = groups705719431365010083at_int(aTP_Lamm_ow(F),A2) ).

% of_int_prod
tff(fact_7272_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A2)) = groups2316167850115554303t_real(aTP_Lamm_gk(F),A2) ).

% of_int_prod
tff(fact_7273_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A2)) = groups1072433553688619179nt_rat(aTP_Lamm_gl(F),A2) ).

% of_int_prod
tff(fact_7274_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_gm(F),A2) ).

% of_int_prod
tff(fact_7275_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( groups6464643781859351333omplex(F,A2) = zero_zero_complex )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_complex(F,X3) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_7276_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups7440179247065528705omplex(F,A2) = zero_zero_complex )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_complex(F,X3) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_7277_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups3708469109370488835omplex(F,A2) = zero_zero_complex )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_complex(F,X3) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_7278_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7279_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups2316167850115554303t_real(F,A2) = zero_zero_real )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7280_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups766887009212190081x_real(F,A2) = zero_zero_real )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7281_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( groups73079841787564623at_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7282_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7283_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7284_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7285_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_7286_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_7287_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_7288_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_7289_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_7290_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_7291_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_7292_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_7293_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_7294_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_7295_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7296_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7297_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7298_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_7299_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_7300_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_7301_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups73079841787564623at_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7302_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7303_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups225925009352817453ex_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7304_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1707563613775114915nt_nat(G,A2) = one_one_nat ) ) ).

% prod.infinite
tff(fact_7305_dvd__prod__eqI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B: nat,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => ( ( B = aa_VEBT_VEBT_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups6361806394783013919BT_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7306_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: nat,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ( ( B = aa_real_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups4696554848551431203al_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7307_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: nat,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ( ( B = aa_int_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups1707563613775114915nt_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7308_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: nat,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => ( ( B = aa_complex_nat(F,A) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),groups861055069439313189ex_nat(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7309_dvd__prod__eqI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B: int,F: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => ( ( B = aa_VEBT_VEBT_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups6359315924273963643BT_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7310_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: int,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ( ( B = aa_real_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups4694064378042380927al_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7311_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: int,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => ( ( B = aa_complex_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups858564598930262913ex_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7312_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: int,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( ( B = aa_nat_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups705719431365010083at_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7313_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ( ( B = aa_int_int(F,A) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),groups1705073143266064639nt_int(F,A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7314_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ( ( 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),A2))) ) ) ) ).

% dvd_prod_eqI
tff(fact_7315_dvd__prodI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A)),groups6361806394783013919BT_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7316_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7317_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7318_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7319_dvd__prodI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_VEBT_VEBT_int(F,A)),groups6359315924273963643BT_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7320_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_real_int(F,A)),groups4694064378042380927al_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7321_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_complex_int(F,A)),groups858564598930262913ex_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7322_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(F,A)),groups705719431365010083at_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7323_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(F,A)),groups1705073143266064639nt_int(F,A2))) ) ) ).

% dvd_prodI
tff(fact_7324_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(F,A)),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ) ).

% dvd_prodI
tff(fact_7325_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_7326_prod_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ox(A),B),S2) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ox(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7327_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_oy(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_oy(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7328_prod_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))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_oz(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_oz(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7329_prod_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))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_pa(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_pa(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7330_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_pb(A),B),S2) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_pb(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7331_prod_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_pc(A),B),S2) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_pc(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7332_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_pd(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pd(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7333_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_pe(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_pe(A),B)),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7334_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_pf(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_pf(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7335_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_pg(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pg(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7336_prod_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ph(A),B),S2) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups127312072573709053omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ph(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7337_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_pi(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_pi(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7338_prod_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))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_pj(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_pj(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7339_prod_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))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_pk(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_pk(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7340_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_pl(A),B),S2) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_pl(A),B),S2) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7341_prod_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_pm(A),B),S2) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_pm(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7342_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_pn(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pn(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7343_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_po(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_po(A),B)),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7344_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_pp(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_pp(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7345_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_pq(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pq(A),B),S2) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7346_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X2)),groups2703838992350267259T_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7347_prod_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups1681761925125756287l_real(G,insert_real(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X2)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7348_prod_Oinsert,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),aa_set_nat_real(groups129246275422532515t_real(G),A2)) ) ) ) ).

% prod.insert
tff(fact_7349_prod_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A2))
       => ( groups2316167850115554303t_real(G,insert_int(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7350_prod_Oinsert,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups766887009212190081x_real(G,insert_complex(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7351_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X2,A2)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X2)),groups5726676334696518183BT_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7352_prod_Oinsert,axiom,
    ! [A2: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups4061424788464935467al_rat(G,insert_real(X2,A2)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X2)),groups4061424788464935467al_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7353_prod_Oinsert,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
       => ( groups73079841787564623at_rat(G,insert_nat(X2,A2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X2)),groups73079841787564623at_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7354_prod_Oinsert,axiom,
    ! [A2: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X2),A2))
       => ( groups1072433553688619179nt_rat(G,insert_int(X2,A2)) = aa_rat_rat(times_times_rat(aa_int_rat(G,X2)),groups1072433553688619179nt_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7355_prod_Oinsert,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups225925009352817453ex_rat(G,insert_complex(X2,A2)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X2)),groups225925009352817453ex_rat(G,A2)) ) ) ) ).

% prod.insert
tff(fact_7356_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_7357_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_7358_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : 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,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% prod.lessThan_Suc
tff(fact_7359_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_7360_and__numerals_I5_J,axiom,
    ! [X2: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit0,X2)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% and_numerals(5)
tff(fact_7361_and__numerals_I5_J,axiom,
    ! [X2: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X2))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_7362_and__numerals_I5_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_7363_and__numerals_I1_J,axiom,
    ! [Y: num] : bit_se3949692690581998587nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,Y))) = zero_z3403309356797280102nteger ).

% and_numerals(1)
tff(fact_7364_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_7365_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_7366_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X2: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_gn(A)),X2))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X2 ) ) ).

% powser_sums_zero_iff
tff(fact_7367_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X2: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_gp(A)),X2))
    <=> ( aa_nat_real(A,zero_zero_nat) = X2 ) ) ).

% powser_sums_zero_iff
tff(fact_7368_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,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)),M))
       => ( groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,set_or1269000886237332187st_nat(M,N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7369_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,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)),M))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7370_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,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)),M))
       => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7371_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,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)),M))
       => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7372_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,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)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7373_and__numerals_I7_J,axiom,
    ! [X2: num,Y: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit1,X2)),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = plus_p5714425477246183910nteger(one_one_Code_integer,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_se3949692690581998587nteger(numera6620942414471956472nteger(X2),numera6620942414471956472nteger(Y)))) ).

% and_numerals(7)
tff(fact_7374_and__numerals_I7_J,axiom,
    ! [X2: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X2))),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,X2)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_7375_and__numerals_I7_J,axiom,
    ! [X2: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2))),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,X2)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_7376_sums__0,axiom,
    ! [F: fun_nat_complex] :
      ( ! [N2: nat] : aa_nat_complex(F,N2) = zero_zero_complex
     => pp(aa_complex_bool(sums_complex(F),zero_zero_complex)) ) ).

% sums_0
tff(fact_7377_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N2: nat] : aa_nat_real(F,N2) = zero_zero_real
     => pp(aa_real_bool(sums_real(F),zero_zero_real)) ) ).

% sums_0
tff(fact_7378_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N2: nat] : aa_nat_nat(F,N2) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F),zero_zero_nat)) ) ).

% sums_0
tff(fact_7379_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N2: nat] : aa_nat_int(F,N2) = zero_zero_int
     => pp(aa_int_bool(sums_int(F),zero_zero_int)) ) ).

% sums_0
tff(fact_7380_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,T: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
     => ( 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_7381_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
     => ( 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_7382_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,T: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
     => ( 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_7383_sums__single,axiom,
    ! [I: nat,F: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ga(I),F)),aa_nat_complex(F,I))) ).

% sums_single
tff(fact_7384_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_pr(I),F)),aa_nat_real(F,I))) ).

% sums_single
tff(fact_7385_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ps(I),F)),aa_nat_nat(F,I))) ).

% sums_single
tff(fact_7386_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_pt(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_7387_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_hk(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% sums_add
tff(fact_7388_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_hi(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% sums_add
tff(fact_7389_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_pu(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% sums_add
tff(fact_7390_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,G: fun_VE7331017413731640256at_int,R: fun_VE3763956218935105199t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups6359315924273963643BT_int(aa_fun6141810409722670662BT_int(aa_fun6474440919679335769BT_int(aTP_Lamm_pv(B2),G),R),A2) = groups705719431365010083at_int(aa_fun1384843155787029862at_int(aa_fun6066863168309489065at_int(aTP_Lamm_px(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7391_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_int,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4694064378042380927al_int(aa_fun1626300625876492518al_int(aa_fun1560246819506852821al_int(aTP_Lamm_py(B2),G),R),A2) = groups705719431365010083at_int(aa_fun1494899438858969354at_int(aa_fun6942549246588167161at_int(aTP_Lamm_qa(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7392_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co7370231399432144932at_int,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups858564598930262913ex_int(aa_fun4872694516007147366ex_int(aa_fun2110580046701903059ex_int(aTP_Lamm_qb(B2),G),R),A2) = groups705719431365010083at_int(aa_fun6041977265744554632at_int(aa_fun3475035881565545205at_int(aTP_Lamm_qd(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7393_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_int,G: fun_VE294566951569886236nt_int,R: fun_VE3303673267288458579t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups6359315924273963643BT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_qe(B2),G),R),A2) = groups1705073143266064639nt_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_qf(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7394_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_int,G: fun_real_fun_int_int,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups4694064378042380927al_int(aa_fun529217645232493962al_int(aa_fun5832317237790237469al_int(aTP_Lamm_qg(B2),G),R),A2) = groups1705073143266064639nt_int(aa_fun5532095235224157706nt_int(aa_fun4786959800906006429nt_int(aTP_Lamm_qh(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7395_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_int,G: fun_co333780937270390912nt_int,R: fun_co2381122952212249071t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B2))
       => ( groups858564598930262913ex_int(aa_fun3700423334941611530ex_int(aa_fun3520494530470439451ex_int(aTP_Lamm_qi(B2),G),R),A2) = groups1705073143266064639nt_int(aa_fun855801025254967176nt_int(aa_fun315996177946036377nt_int(aTP_Lamm_qj(A2),G),R),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7396_prod_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B2: set_nat,G: fun_VE597728521808591204at_nat,R: fun_VE3763956218935105199t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups6361806394783013919BT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_qk(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_ql(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7397_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups4696554848551431203al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_qm(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_qn(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7398_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups1707563613775114915nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_qo(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_qp(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7399_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( groups861055069439313189ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_qq(B2),G),R),A2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_qr(A2),G),R)),B2) ) ) ) ).

% prod.swap_restrict
tff(fact_7400_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A2))
         => ( 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),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2))) ) ).

% prod_mono
tff(fact_7401_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I3)),aa_VEBT_VEBT_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(G,A2))) ) ).

% prod_mono
tff(fact_7402_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A2))
         => ( 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,A2)),groups2316167850115554303t_real(G,A2))) ) ).

% prod_mono
tff(fact_7403_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A2))
         => ( 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,A2)),groups1681761925125756287l_real(G,A2))) ) ).

% prod_mono
tff(fact_7404_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A2))
         => ( 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,A2)),groups73079841787564623at_rat(G,A2))) ) ).

% prod_mono
tff(fact_7405_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(G,A2))) ) ).

% prod_mono
tff(fact_7406_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A2))
         => ( 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,A2)),groups1072433553688619179nt_rat(G,A2))) ) ).

% prod_mono
tff(fact_7407_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A2))
         => ( 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,A2)),groups4061424788464935467al_rat(G,A2))) ) ).

% prod_mono
tff(fact_7408_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I3)),aa_VEBT_VEBT_nat(G,I3))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups6361806394783013919BT_nat(F,A2)),groups6361806394783013919BT_nat(G,A2))) ) ).

% prod_mono
tff(fact_7409_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A2))
         => ( 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,A2)),groups1707563613775114915nt_nat(G,A2))) ) ).

% prod_mono
tff(fact_7410_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),groups705719431365010083at_int(F,A2))) ) ).

% prod_nonneg
tff(fact_7411_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% prod_nonneg
tff(fact_7412_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),A2))) ) ).

% prod_nonneg
tff(fact_7413_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),groups705719431365010083at_int(F,A2))) ) ).

% prod_pos
tff(fact_7414_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% prod_pos
tff(fact_7415_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),A2))) ) ).

% prod_pos
tff(fact_7416_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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),A2))) ) ).

% prod_ge_1
tff(fact_7417_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2703838992350267259T_real(F,A2))) ) ).

% prod_ge_1
tff(fact_7418_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_7419_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_7420_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_7421_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups5726676334696518183BT_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_7422_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_7423_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_7424_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups6361806394783013919BT_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_7425_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => 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,A2))) ) ).

% prod_ge_1
tff(fact_7426_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & ( aa_nat_complex(F,X) = zero_zero_complex ) )
       => ( groups6464643781859351333omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7427_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & ( aa_int_complex(F,X) = zero_zero_complex ) )
       => ( groups7440179247065528705omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7428_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X: complex] :
            ( pp(aa_set_complex_bool(member_complex(X),A2))
            & ( aa_complex_complex(F,X) = zero_zero_complex ) )
       => ( groups3708469109370488835omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_7429_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & ( aa_nat_real(F,X) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7430_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & ( aa_int_real(F,X) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7431_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X: complex] :
            ( pp(aa_set_complex_bool(member_complex(X),A2))
            & ( aa_complex_real(F,X) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7432_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & ( aa_nat_rat(F,X) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7433_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & ( aa_int_rat(F,X) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7434_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X: complex] :
            ( pp(aa_set_complex_bool(member_complex(X),A2))
            & ( aa_complex_rat(F,X) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7435_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & ( aa_int_nat(F,X) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_7436_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups6464643781859351333omplex(F,set_or1269000886237332187st_nat(A,B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_qs(F),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_7437_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups129246275422532515t_real(F),set_or1269000886237332187st_nat(A,B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_qt(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_7438_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,set_or1269000886237332187st_nat(A,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_qu(F),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_7439_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,set_or1269000886237332187st_nat(A,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_qv(F),A),B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_7440_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_qw(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_7441_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qx(C),F)),aa_complex_complex(times_times_complex(C),D)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult_iff
tff(fact_7442_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qy(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult_iff
tff(fact_7443_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qz(C),F)),aa_complex_complex(times_times_complex(D),C)))
      <=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_7444_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ra(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_7445_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_7446_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_7447_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( groups127312072573709053omplex(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(A2),P))) = groups127312072573709053omplex(aa_fun6792494735270171281omplex(aTP_Lamm_rb(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7448_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups713298508707869441omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(A2),P))) = groups713298508707869441omplex(aa_fun6813637976755515871omplex(aTP_Lamm_rc(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7449_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bf(A2),P))) = groups6464643781859351333omplex(aa_fun8588439219086927967omplex(aTP_Lamm_rd(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7450_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(A2),P))) = groups7440179247065528705omplex(aa_fun3606824691168040799omplex(aTP_Lamm_re(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7451_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kc(A2),P))) = groups3708469109370488835omplex(aa_fun1034237174562290143omplex(aTP_Lamm_rf(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7452_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( groups2703838992350267259T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(A2),P))) = groups2703838992350267259T_real(aa_fun2072160908118572431T_real(aTP_Lamm_rg(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7453_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(A2),P))) = groups1681761925125756287l_real(aa_fun6348708079608492509l_real(aTP_Lamm_rh(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7454_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_real,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bf(A2),P))) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1750223370215434333t_real(aTP_Lamm_ri(G),P)),A2) ) ) ).

% prod.inter_filter
tff(fact_7455_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(A2),P))) = groups2316167850115554303t_real(aa_fun6974987029372589917t_real(aTP_Lamm_rj(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7456_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kc(A2),P))) = groups766887009212190081x_real(aa_fun2902006202505214429x_real(aTP_Lamm_rk(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7457_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = groups705719431365010083at_int(aTP_Lamm_lg(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7458_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kl(G)),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7459_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_rl(G),K),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7460_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kn(G),K)),set_or1269000886237332187st_nat(M,N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7461_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( 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),A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7462_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7463_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( 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,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7464_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( 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,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7465_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( 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,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7466_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7467_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( 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,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7468_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => ( 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,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7469_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups6361806394783013919BT_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7470_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( 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,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7471_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))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),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_7472_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))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),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_7473_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))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),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_7474_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))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),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_7475_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))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),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_7476_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))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),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_7477_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))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),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_7478_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))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),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_7479_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))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),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_7480_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))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),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_7481_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X2,A2)) = groups2703838992350267259T_real(G,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X2)),groups2703838992350267259T_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7482_prod_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups1681761925125756287l_real(G,insert_real(X2,A2)) = groups1681761925125756287l_real(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups1681761925125756287l_real(G,insert_real(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X2)),groups1681761925125756287l_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7483_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X2,A2)) = aa_set_nat_real(groups129246275422532515t_real(G),A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),aa_set_nat_real(groups129246275422532515t_real(G),A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7484_prod_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups2316167850115554303t_real(G,insert_int(X2,A2)) = groups2316167850115554303t_real(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups2316167850115554303t_real(G,insert_int(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7485_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups766887009212190081x_real(G,insert_complex(X2,A2)) = groups766887009212190081x_real(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups766887009212190081x_real(G,insert_complex(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7486_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X2,A2)) = groups5726676334696518183BT_rat(G,A2) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
         => ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X2,A2)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X2)),groups5726676334696518183BT_rat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7487_prod_Oinsert__if,axiom,
    ! [A2: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups4061424788464935467al_rat(G,insert_real(X2,A2)) = groups4061424788464935467al_rat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X2),A2))
         => ( groups4061424788464935467al_rat(G,insert_real(X2,A2)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X2)),groups4061424788464935467al_rat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7488_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( groups73079841787564623at_rat(G,insert_nat(X2,A2)) = groups73079841787564623at_rat(G,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( groups73079841787564623at_rat(G,insert_nat(X2,A2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X2)),groups73079841787564623at_rat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7489_prod_Oinsert__if,axiom,
    ! [A2: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups1072433553688619179nt_rat(G,insert_int(X2,A2)) = groups1072433553688619179nt_rat(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X2),A2))
         => ( groups1072433553688619179nt_rat(G,insert_int(X2,A2)) = aa_rat_rat(times_times_rat(aa_int_rat(G,X2)),groups1072433553688619179nt_rat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7490_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups225925009352817453ex_rat(G,insert_complex(X2,A2)) = groups225925009352817453ex_rat(G,A2) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X2),A2))
         => ( groups225925009352817453ex_rat(G,insert_complex(X2,A2)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X2)),groups225925009352817453ex_rat(G,A2)) ) ) ) ) ).

% prod.insert_if
tff(fact_7491_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7492_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7493_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7494_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups705719431365010083at_int(F,A2)),groups705719431365010083at_int(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7495_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(F,B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7496_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(F),B2))) ) ) ).

% prod_dvd_prod_subset
tff(fact_7497_prod__dvd__prod__subset2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A3)),aa_VEBT_VEBT_nat(G,A3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,groups6361806394783013919BT_nat(F,A2)),groups6361806394783013919BT_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7498_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => 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,A2)),groups4696554848551431203al_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7499_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A2))
             => 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,A2)),groups861055069439313189ex_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7500_prod__dvd__prod__subset2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_VEBT_VEBT_int(F,A3)),aa_VEBT_VEBT_int(G,A3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,groups6359315924273963643BT_int(F,A2)),groups6359315924273963643BT_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7501_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => 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,A2)),groups4694064378042380927al_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7502_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),A2))
             => 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,A2)),groups858564598930262913ex_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7503_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A2))
             => 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,A2)),groups1707563613775114915nt_nat(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7504_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A2))
             => 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,groups705719431365010083at_int(F,A2)),groups705719431365010083at_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7505_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A2))
             => 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,A2)),groups1705073143266064639nt_int(G,B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7506_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A2))
             => 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),A2)),aa_set_nat_nat(groups708209901874060359at_nat(G),B2))) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_7507_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T3: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T3,T4))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),T4))
                       => ( aa_VEBT_VEBT_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups127312072573709053omplex(G,S2) = groups127312072573709053omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7508_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_real,S2: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T3: set_real,G: fun_VE5569070535936029429omplex,H: fun_real_complex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(J,A3)),minus_minus_set_real(T3,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_real_complex(H,aa_VEBT_VEBT_real(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups127312072573709053omplex(G,S2) = groups713298508707869441omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7509_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_VEBT_VEBT,S2: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_real_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_real_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T3,T4))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                   => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(I,B3)),minus_minus_set_real(S2,S4))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S4))
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),T4))
                       => ( aa_VEBT_VEBT_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_real_VEBT_VEBT(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups127312072573709053omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7510_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J,A3)),minus_minus_set_real(T3,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S4))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S4))
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_complex(H,B3) = one_one_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) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7511_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_int,S2: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T3: set_int,G: fun_VE5569070535936029429omplex,H: fun_int_complex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(J,A3)),minus_minus_set_int(T3,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_int_complex(H,aa_VEBT_VEBT_int(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups127312072573709053omplex(G,S2) = groups7440179247065528705omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7512_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T3: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T3,T4))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T3,T4)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S4))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S4))
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T4))
                       => ( aa_int_complex(H,B3) = one_one_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) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups7440179247065528705omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7513_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_complex,S2: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T3: set_complex,G: fun_VE5569070535936029429omplex,H: fun_complex_complex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [A3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(S2,S4)))
               => pp(aa_set_complex_bool(member_complex(aa_VEBT_VEBT_complex(J,A3)),minus_811609699411566653omplex(T3,T4))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T4)))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T4)))
                   => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_complex_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S4))) )
               => ( ! [A3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S4))
                     => ( aa_VEBT_VEBT_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T4))
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),S2))
                         => ( aa_complex_complex(H,aa_VEBT_VEBT_complex(J,A3)) = aa_VEBT_VEBT_complex(G,A3) ) )
                     => ( groups127312072573709053omplex(G,S2) = groups3708469109370488835omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7514_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T3: set_complex,G: fun_real_complex,H: fun_complex_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S2,S4)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T3,T4))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T4)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T3,T4)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S2,S4))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S4))
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T4))
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S2))
                         => ( aa_complex_complex(H,aa_real_complex(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups3708469109370488835omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7515_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_VEBT_VEBT,S2: set_int,I: fun_VEBT_VEBT_int,J: fun_int_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_int_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
             => ( aa_VEBT_VEBT_int(I,aa_int_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
               => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,aa_int_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T3,T4))) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                 => ( aa_int_VEBT_VEBT(J,aa_VEBT_VEBT_int(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(T3,T4)))
                   => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),T4))
                       => ( aa_VEBT_VEBT_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_int_VEBT_VEBT(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups127312072573709053omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7516_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T3: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J,A3)),minus_minus_set_real(T3,T4))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T3,T4)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T4))
                       => ( aa_real_complex(H,B3) = one_one_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) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups713298508707869441omplex(H,T3) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7517_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qx(C),F)),A))
     => ( ( C != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ) ).

% sums_mult_D
tff(fact_7518_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qy(C),F)),A))
     => ( ( C != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).

% sums_mult_D
tff(fact_7519_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_rm(F)),S))
       => pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_7520_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_km(F)),S))
       => pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_7521_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_km(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_7522_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_km(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_7523_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_kl(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_7524_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_lg(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_7525_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X2: 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(X2),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_7526_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_7527_and__less__eq,axiom,
    ! [L: int,K: 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(K),L)),K)) ) ).

% and_less_eq
tff(fact_7528_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_rn(N),F)),S))
      <=> pp(aa_complex_bool(sums_complex(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_7529_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_ro(N),F)),S))
      <=> pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_7530_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => pp(aa_complex_bool(sums_complex(F),groups2073611262835488442omplex(F,N4))) ) ) ).

% sums_finite
tff(fact_7531_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),groups3539618377306564664at_int(F,N4))) ) ) ).

% sums_finite
tff(fact_7532_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N4))) ) ) ).

% sums_finite
tff(fact_7533_sums__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N4))) ) ) ).

% sums_finite
tff(fact_7534_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rp(P),F)),groups2073611262835488442omplex(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7535_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_rq(P),F)),groups3539618377306564664at_int(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7536_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rr(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7537_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_rs(P),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_7538_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rt(A2),F)),groups2073611262835488442omplex(F,A2))) ) ).

% sums_If_finite_set
tff(fact_7539_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ru(A2),F)),groups3539618377306564664at_int(F,A2))) ) ).

% sums_If_finite_set
tff(fact_7540_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rv(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2))) ) ).

% sums_If_finite_set
tff(fact_7541_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_rw(A2),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A2))) ) ).

% sums_If_finite_set
tff(fact_7542_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_rx(G)))) = groups713298508707869441omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7543_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ry(G)))) = groups7440179247065528705omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7544_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_rz(G)))) = groups3708469109370488835omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7545_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1681761925125756287l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_sa(G)))) = groups1681761925125756287l_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7546_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_sb(G)))) = groups2316167850115554303t_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7547_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_sc(G)))) = groups766887009212190081x_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7548_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4061424788464935467al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_sd(G)))) = groups4061424788464935467al_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7549_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_se(G)))) = groups1072433553688619179nt_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7550_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_sf(G)))) = groups225925009352817453ex_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7551_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4696554848551431203al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_sg(G)))) = groups4696554848551431203al_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7552_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sh(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_7553_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ng(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_7554_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N: nat,M: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(N,M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_si(G),N),M),set_or1269000886237332187st_nat(N,M)) ).

% prod.atLeastAtMost_rev
tff(fact_7555_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(N,M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ld(G),N),M)),set_or1269000886237332187st_nat(N,M)) ).

% prod.atLeastAtMost_rev
tff(fact_7556_less__1__prod2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7557_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I)))
         => ( ! [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_7558_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I)))
         => ( ! [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_7559_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I)))
         => ( ! [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_7560_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I)))
         => ( ! [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_7561_less__1__prod2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I6))) ) ) ) ) ).

% less_1_prod2
tff(fact_7562_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I6))
     => ( pp(aa_set_real_bool(member_real(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I)))
         => ( ! [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_7563_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I6))
     => ( pp(aa_set_nat_bool(member_nat(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I)))
         => ( ! [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_7564_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I6))
     => ( pp(aa_set_int_bool(member_int(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I)))
         => ( ! [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_7565_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I6))
     => ( pp(aa_set_complex_bool(member_complex(I),I6))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I)))
         => ( ! [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_7566_less__1__prod,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7567_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_7568_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_7569_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_7570_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_7571_less__1__prod,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I6))) ) ) ) ).

% less_1_prod
tff(fact_7572_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_7573_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_7574_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_7575_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_7576_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,B2))),groups766887009212190081x_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7577_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A2,B2))),aa_set_nat_real(groups129246275422532515t_real(G),B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7578_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups225925009352817453ex_rat(G,A2) = aa_rat_rat(times_times_rat(groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups225925009352817453ex_rat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7579_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups73079841787564623at_rat(G,A2) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,minus_minus_set_nat(A2,B2))),groups73079841787564623at_rat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7580_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups861055069439313189ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups861055069439313189ex_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7581_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups858564598930262913ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,B2))),groups858564598930262913ex_int(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7582_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups2316167850115554303t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G,minus_minus_set_int(A2,B2))),groups2316167850115554303t_real(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7583_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups1072433553688619179nt_rat(G,A2) = aa_rat_rat(times_times_rat(groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,B2))),groups1072433553688619179nt_rat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7584_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,B2))),groups1707563613775114915nt_nat(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7585_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups705719431365010083at_int(G,A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,minus_minus_set_nat(A2,B2))),groups705719431365010083at_int(G,B2)) ) ) ) ).

% prod.subset_diff
tff(fact_7586_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_complex(G,X4) = aa_VEBT_VEBT_complex(H,X4) ) )
           => ( groups127312072573709053omplex(G,T3) = groups127312072573709053omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7587_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) = groups713298508707869441omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7588_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) = groups3708469109370488835omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7589_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_real(G,X4) = one_one_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2703838992350267259T_real(G,T3) = groups2703838992350267259T_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7590_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) = groups1681761925125756287l_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7591_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) = groups766887009212190081x_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7592_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups5726676334696518183BT_rat(G,T3) = groups5726676334696518183BT_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7593_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) = groups4061424788464935467al_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7594_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) = groups225925009352817453ex_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7595_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_nat(G,X4) = aa_VEBT_VEBT_nat(H,X4) ) )
           => ( groups6361806394783013919BT_nat(G,T3) = groups6361806394783013919BT_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7596_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VE5569070535936029429omplex,G: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_complex(G,X4) = aa_VEBT_VEBT_complex(H,X4) ) )
           => ( groups127312072573709053omplex(G,S2) = groups127312072573709053omplex(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7597_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7598_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7599_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_real(H,X4) = one_one_real ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_real(G,X4) = aa_VEBT_VEBT_real(H,X4) ) )
           => ( groups2703838992350267259T_real(G,S2) = groups2703838992350267259T_real(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7600_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7601_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7602_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_rat(G,X4) = aa_VEBT_VEBT_rat(H,X4) ) )
           => ( groups5726676334696518183BT_rat(G,S2) = groups5726676334696518183BT_rat(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7603_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T3))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T3,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,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7604_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,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,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7605_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,T3))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S2),T3))
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(T3,S2)))
             => ( aa_VEBT_VEBT_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),S2))
               => ( aa_VEBT_VEBT_nat(G,X4) = aa_VEBT_VEBT_nat(H,X4) ) )
           => ( groups6361806394783013919BT_nat(G,S2) = groups6361806394783013919BT_nat(H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7606_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T3) = groups3708469109370488835omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7607_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T3) = groups766887009212190081x_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7608_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T3) = groups225925009352817453ex_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7609_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T3) = groups861055069439313189ex_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7610_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T3) = groups858564598930262913ex_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7611_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,T3) = groups6464643781859351333omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7612_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),T3) = aa_set_nat_real(groups129246275422532515t_real(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7613_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,T3) = groups73079841787564623at_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7614_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,T3) = groups7440179247065528705omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7615_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,T3) = groups2316167850115554303t_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7616_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7617_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7618_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7619_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7620_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T3))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T3,S2)))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7621_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,S2) = groups6464643781859351333omplex(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7622_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),S2) = aa_set_nat_real(groups129246275422532515t_real(G),T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7623_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),T3))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(T3,S2)))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,S2) = groups73079841787564623at_rat(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7624_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,S2) = groups7440179247065528705omplex(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7625_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T3))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T3,S2)))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,S2) = groups2316167850115554303t_real(G,T3) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7626_prod_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_complex(H,B3) = one_one_complex ) )
             => ( ( groups127312072573709053omplex(G,C4) = groups127312072573709053omplex(H,C4) )
               => ( groups127312072573709053omplex(G,A2) = groups127312072573709053omplex(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7627_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) )
               => ( groups713298508707869441omplex(G,A2) = groups713298508707869441omplex(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7628_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) )
               => ( groups3708469109370488835omplex(G,A2) = groups3708469109370488835omplex(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7629_prod_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_real(H,B3) = one_one_real ) )
             => ( ( groups2703838992350267259T_real(G,C4) = groups2703838992350267259T_real(H,C4) )
               => ( groups2703838992350267259T_real(G,A2) = groups2703838992350267259T_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7630_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
               => ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7631_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) )
               => ( groups766887009212190081x_real(G,A2) = groups766887009212190081x_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7632_prod_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = one_one_rat ) )
             => ( ( groups5726676334696518183BT_rat(G,C4) = groups5726676334696518183BT_rat(H,C4) )
               => ( groups5726676334696518183BT_rat(G,A2) = groups5726676334696518183BT_rat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7633_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) )
               => ( groups4061424788464935467al_rat(G,A2) = groups4061424788464935467al_rat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7634_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) )
               => ( groups225925009352817453ex_rat(G,A2) = groups225925009352817453ex_rat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7635_prod_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_nat(H,B3) = one_one_nat ) )
             => ( ( groups6361806394783013919BT_nat(G,C4) = groups6361806394783013919BT_nat(H,C4) )
               => ( groups6361806394783013919BT_nat(G,A2) = groups6361806394783013919BT_nat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7636_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_complex(H,B3) = one_one_complex ) )
             => ( ( groups127312072573709053omplex(G,A2) = groups127312072573709053omplex(H,B2) )
              <=> ( groups127312072573709053omplex(G,C4) = groups127312072573709053omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7637_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A2) = groups713298508707869441omplex(H,B2) )
              <=> ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7638_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A2) = groups3708469109370488835omplex(H,B2) )
              <=> ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7639_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_real(G,A3) = one_one_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_real(H,B3) = one_one_real ) )
             => ( ( groups2703838992350267259T_real(G,A2) = groups2703838992350267259T_real(H,B2) )
              <=> ( groups2703838992350267259T_real(G,C4) = groups2703838992350267259T_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7640_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B2) )
              <=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7641_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A2) = groups766887009212190081x_real(H,B2) )
              <=> ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7642_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_rat(H,B3) = one_one_rat ) )
             => ( ( groups5726676334696518183BT_rat(G,A2) = groups5726676334696518183BT_rat(H,B2) )
              <=> ( groups5726676334696518183BT_rat(G,C4) = groups5726676334696518183BT_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7643_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: 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(A2),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A2)))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B2)))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A2) = groups4061424788464935467al_rat(H,B2) )
              <=> ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7644_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,A2) = groups225925009352817453ex_rat(H,B2) )
              <=> ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7645_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,C4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),C4))
       => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B2),C4))
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),minus_5127226145743854075T_VEBT(C4,A2)))
               => ( aa_VEBT_VEBT_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(C4,B2)))
                 => ( aa_VEBT_VEBT_nat(H,B3) = one_one_nat ) )
             => ( ( groups6361806394783013919BT_nat(G,A2) = groups6361806394783013919BT_nat(H,B2) )
              <=> ( groups6361806394783013919BT_nat(G,C4) = groups6361806394783013919BT_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7646_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_sj(M),Z)),aa_nat_complex(power_power_complex(Z),M))) ).

% powser_sums_if
tff(fact_7647_powser__sums__if,axiom,
    ! [M: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_sk(M),Z)),aa_nat_real(power_power_real(Z),M))) ).

% powser_sums_if
tff(fact_7648_powser__sums__if,axiom,
    ! [M: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_sl(M),Z)),aa_nat_int(power_power_int(Z),M))) ).

% powser_sums_if
tff(fact_7649_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),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),set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7650_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7651_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7652_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),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),set_or1269000886237332187st_nat(zero_zero_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7653_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_gn(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7654_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_gp(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_7655_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,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),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7656_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7657_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,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))),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7658_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,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),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7659_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7660_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7661_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7662_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7663_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_ko(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_7664_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_ko(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_7665_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_ko(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_7666_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A2: set_nat,S4: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G),S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( ( S4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sm(G),F)),A2)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_sn(G),A2),F)),S4)) ) ) ) ).

% sums_If_finite_set'
tff(fact_7667_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_7668_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_km(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7669_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_lf(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7670_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : 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)),groups705719431365010083at_int(aTP_Lamm_lg(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7671_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_kl(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_7672_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_km(G)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7673_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(aTP_Lamm_lf(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7674_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,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,M)),groups705719431365010083at_int(aTP_Lamm_lg(G),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7675_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,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,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kl(G)),set_or1269000886237332187st_nat(M,N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7676_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = groups705719431365010083at_int(aTP_Lamm_lg(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_7677_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kl(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_7678_prod__mono__strict,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_VEBT_VEBT_real(F,I3)),aa_VEBT_VEBT_real(G,I3))) ) )
       => ( ( A2 != bot_bo8194388402131092736T_VEBT )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7679_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7680_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7681_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7682_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7683_prod__mono__strict,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),A2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3))) ) )
       => ( ( A2 != bot_bo8194388402131092736T_VEBT )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7684_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7685_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7686_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_nat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups73079841787564623at_rat(F,A2)),groups73079841787564623at_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7687_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A2))
           => ( 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))) ) )
       => ( ( A2 != bot_bot_set_int )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7688_even__prod__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( 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,A2)))
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & 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,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7689_even__prod__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( 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,A2)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & 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,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7690_even__prod__iff,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( 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,A2)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & 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,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7691_even__prod__iff,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3455450783089532116nteger(F,A2)))
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7692_even__prod__iff,axiom,
    ! [A2: set_int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3827104343326376752nteger(F,A2)))
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7693_even__prod__iff,axiom,
    ! [A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups8682486955453173170nteger(F,A2)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_com4217951221296830281nteger(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7694_even__prod__iff,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups705719431365010083at_int(F,A2)))
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & 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,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7695_even__prod__iff,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( 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,A2)))
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & 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,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7696_even__prod__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( 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),A2)))
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & 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,X3))) ) ) ) ).

% even_prod_iff
tff(fact_7697_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,insert_complex(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_7698_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_real,X2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1681761925125756287l_real(G,insert_real(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X2)),groups1681761925125756287l_real(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_7699_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_real,X2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,insert_int(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_7700_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_real,X2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),insert_nat(X2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A2,insert_nat(X2,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_7701_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_rat,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups225925009352817453ex_rat(G,insert_complex(X2,A2)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X2)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_7702_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_rat,X2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4061424788464935467al_rat(G,insert_real(X2,A2)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X2)),groups4061424788464935467al_rat(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_7703_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_rat,X2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1072433553688619179nt_rat(G,insert_int(X2,A2)) = aa_rat_rat(times_times_rat(aa_int_rat(G,X2)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_7704_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_rat,X2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups73079841787564623at_rat(G,insert_nat(X2,A2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X2)),groups73079841787564623at_rat(G,minus_minus_set_nat(A2,insert_nat(X2,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_7705_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups861055069439313189ex_nat(G,insert_complex(X2,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G,X2)),groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_7706_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,X2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups4696554848551431203al_nat(G,insert_real(X2,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G,X2)),groups4696554848551431203al_nat(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_7707_prod_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups2703838992350267259T_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X2)),groups2703838992350267259T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% prod.remove
tff(fact_7708_prod_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X2)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_7709_prod_Oremove,axiom,
    ! [A2: set_real,X2: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups1681761925125756287l_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X2)),groups1681761925125756287l_real(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_7710_prod_Oremove,axiom,
    ! [A2: set_int,X2: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X2),A2))
       => ( groups2316167850115554303t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X2)),groups2316167850115554303t_real(G,minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_7711_prod_Oremove,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(X2),A2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X2)),aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A2,insert_nat(X2,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_7712_prod_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X2: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),A2))
       => ( groups5726676334696518183BT_rat(G,A2) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X2)),groups5726676334696518183BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X2,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% prod.remove
tff(fact_7713_prod_Oremove,axiom,
    ! [A2: set_complex,X2: complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(member_complex(X2),A2))
       => ( groups225925009352817453ex_rat(G,A2) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X2)),groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X2,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_7714_prod_Oremove,axiom,
    ! [A2: set_real,X2: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X2),A2))
       => ( groups4061424788464935467al_rat(G,A2) = aa_rat_rat(times_times_rat(aa_real_rat(G,X2)),groups4061424788464935467al_rat(G,minus_minus_set_real(A2,insert_real(X2,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_7715_prod_Oremove,axiom,
    ! [A2: set_int,X2: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X2),A2))
       => ( groups1072433553688619179nt_rat(G,A2) = aa_rat_rat(times_times_rat(aa_int_rat(G,X2)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,insert_int(X2,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_7716_prod_Oremove,axiom,
    ! [A2: set_nat,X2: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(X2),A2))
       => ( groups73079841787564623at_rat(G,A2) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X2)),groups73079841787564623at_rat(G,minus_minus_set_nat(A2,insert_nat(X2,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_7717_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,N))),aa_set_nat_real(groups129246275422532515t_real(G),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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7718_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,N))),groups73079841787564623at_rat(G,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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7719_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,N))),groups705719431365010083at_int(G,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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7720_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,N))),aa_set_nat_nat(groups708209901874060359at_nat(G),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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7721_prod_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_jl(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(B,A)),groups2703838992350267259T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_jl(A),B),C),S2) = groups2703838992350267259T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7722_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: 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_jm(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(B,A)),groups766887009212190081x_real(C,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_jm(A),B),C),S2) = groups766887009212190081x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7723_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C: 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_jt(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(B,A)),groups1681761925125756287l_real(C,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_jt(A),B),C),S2) = groups1681761925125756287l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7724_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_real,C: 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_so(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(B,A)),groups2316167850115554303t_real(C,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_so(A),B),C),S2) = groups2316167850115554303t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7725_prod_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_real,C: 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_sp(A),B),C)),S2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(B,A)),aa_set_nat_real(groups129246275422532515t_real(C),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_sp(A),B),C)),S2) = aa_set_nat_real(groups129246275422532515t_real(C),minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7726_prod_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
     => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups5726676334696518183BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_jn(A),B),C),S2) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(B,A)),groups5726676334696518183BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
         => ( groups5726676334696518183BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_jn(A),B),C),S2) = groups5726676334696518183BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7727_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: 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_jo(A),B),C),S2) = aa_rat_rat(times_times_rat(aa_complex_rat(B,A)),groups225925009352817453ex_rat(C,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_jo(A),B),C),S2) = groups225925009352817453ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7728_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C: 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_ju(A),B),C),S2) = aa_rat_rat(times_times_rat(aa_real_rat(B,A)),groups4061424788464935467al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_ju(A),B),C),S2) = groups4061424788464935467al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7729_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_sq(A),B),C),S2) = aa_rat_rat(times_times_rat(aa_int_rat(B,A)),groups1072433553688619179nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_sq(A),B),C),S2) = groups1072433553688619179nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7730_prod_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat,C: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sr(A),B),C),S2) = aa_rat_rat(times_times_rat(aa_nat_rat(B,A)),groups73079841787564623at_rat(C,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat)))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sr(A),B),C),S2) = groups73079841787564623at_rat(C,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_7731_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X2: fun_nat_fun_nat_nat,Xa2: nat,Xb3: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X2,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(X2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb3),aa_nat_nat(aa_nat_fun_nat_nat(X2,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_7732_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_7733_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_7734_prod__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,B3))) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7735_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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),A2))
               => 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,A2)),groups1681761925125756287l_real(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7736_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => 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),A2))
               => 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,A2)),groups766887009212190081x_real(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7737_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2)))
             => 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),A2))
               => 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),A2)),aa_set_nat_real(groups129246275422532515t_real(F),B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7738_prod__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,B3))) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7739_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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),A2))
               => 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,A2)),groups4061424788464935467al_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7740_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => 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),A2))
               => 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,A2)),groups225925009352817453ex_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7741_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B2))
       => ( ! [B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2)))
             => 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),A2))
               => 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,A2)),groups73079841787564623at_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7742_prod__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2))
       => ( ! [B3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,B3),minus_5127226145743854075T_VEBT(B2,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_VEBT_VEBT_int(F,B3))) )
         => ( ! [A3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A3),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_VEBT_VEBT_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups6359315924273963643BT_int(F,A2)),groups6359315924273963643BT_int(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7743_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B2,A2)))
             => 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),A2))
               => 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,A2)),groups4694064378042380927al_int(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7744_prod__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( aa_VEBT_VEBT_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
           => ( groups127312072573709053omplex(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_complex_complex(divide1717551699836669952omplex(groups127312072573709053omplex(F,A2)),aa_VEBT_VEBT_complex(F,A)) ) )
          & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
           => ( groups127312072573709053omplex(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups127312072573709053omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7745_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( aa_complex_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F,A2)),aa_complex_complex(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups3708469109370488835omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7746_prod__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ( aa_VEBT_VEBT_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
           => ( groups5726676334696518183BT_rat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_rat_rat(divide_divide_rat(groups5726676334696518183BT_rat(F,A2)),aa_VEBT_VEBT_rat(F,A)) ) )
          & ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
           => ( groups5726676334696518183BT_rat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups5726676334696518183BT_rat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7747_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( aa_complex_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups225925009352817453ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_rat_rat(divide_divide_rat(groups225925009352817453ex_rat(F,A2)),aa_complex_rat(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
           => ( groups225925009352817453ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups225925009352817453ex_rat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7748_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_complex,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( aa_real_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_real_bool(member_real(A),A2))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F,A2)),aa_real_complex(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A2))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups713298508707869441omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7749_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_rat,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( aa_real_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_real_bool(member_real(A),A2))
           => ( groups4061424788464935467al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_rat_rat(divide_divide_rat(groups4061424788464935467al_rat(F,A2)),aa_real_rat(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A2))
           => ( groups4061424788464935467al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups4061424788464935467al_rat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7750_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_complex,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( aa_int_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_int_bool(member_int(A),A2))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F,A2)),aa_int_complex(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A2))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups7440179247065528705omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7751_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_rat,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( aa_int_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_int_bool(member_int(A),A2))
           => ( groups1072433553688619179nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_rat_rat(divide_divide_rat(groups1072433553688619179nt_rat(F,A2)),aa_int_rat(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A2))
           => ( groups1072433553688619179nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups1072433553688619179nt_rat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7752_prod__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_complex,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_nat_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A2))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(F,A2)),aa_nat_complex(F,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
           => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = groups6464643781859351333omplex(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7753_prod__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_nat_rat(F,A) != zero_zero_rat )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A2))
           => ( groups73079841787564623at_rat(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(F,A2)),aa_nat_rat(F,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
           => ( groups73079841787564623at_rat(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = groups73079841787564623at_rat(F,A2) ) ) ) ) ) ).

% prod_diff1
tff(fact_7754_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aTP_Lamm_ss(A),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_7755_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_st(A)),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_7756_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aTP_Lamm_su(A),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_7757_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_sv,A)),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod
tff(fact_7758_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_7759_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_sw(A),N),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_7760_pochhammer__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),N) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_sx(A),N)),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_7761_pochhammer__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sy(A),N),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_7762_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_sz(A),N)),set_or1269000886237332187st_nat(one_one_nat,N)) ).

% pochhammer_prod_rev
tff(fact_7763_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_7764_geometric__sums,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => pp(aa_real_bool(sums_real(power_power_real(C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)))) ) ).

% geometric_sums
tff(fact_7765_geometric__sums,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => pp(aa_complex_bool(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)))) ) ).

% geometric_sums
tff(fact_7766_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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_ta(G)),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_7767_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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_tb(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_7768_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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)))) = groups705719431365010083at_int(aTP_Lamm_tc(G),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_7769_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),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_td(G)),set_or1269000886237332187st_nat(M,N)) ).

% prod.in_pairs
tff(fact_7770_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(A,B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_te(F),A,B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_7771_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(A,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_tf(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7772_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(A,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_tg(F),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7773_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_th(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7774_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(A,B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ti(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_7775_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_sw(A),N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7776_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_sx(A),N)),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7777_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sy(A),N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7778_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_sz(A),N)),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7779_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_7780_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K,L))
     => ( ( ( 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(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(K),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))),K))),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(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(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(K),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))),K))),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(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_7781_and__int_Opelims,axiom,
    ! [X2: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X2),Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X2,Xa2))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X2),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))),X2))),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(X2),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))),X2))),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(X2),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)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X2,Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_7782_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(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_7783_cot__less__zero,axiom,
    ! [X2: 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)))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X2)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_7784_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: 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,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat))))))
       => 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,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_7785_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_complex,X2: 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_tj(A),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X2),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_tl(N),A),X2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7786_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_rat,X2: 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_tm(A),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tm(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_to(N),A),X2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7787_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_int,X2: 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,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(A),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(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,X2),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_tr(N),A),X2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7788_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_real,X2: 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_ts(A),X2)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(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,X2),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_tu(N),A),X2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7789_atMost__eq__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_atMost_nat,X2) = aa_nat_set_nat(set_ord_atMost_nat,Y) )
    <=> ( X2 = Y ) ) ).

% atMost_eq_iff
tff(fact_7790_atMost__eq__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( set_ord_atMost_int(X2) = set_ord_atMost_int(Y) )
    <=> ( X2 = Y ) ) ).

% atMost_eq_iff
tff(fact_7791_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or4236626031148496127et_nat(K)))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I),K)) ) ).

% atMost_iff
tff(fact_7792_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_ord_atMost_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),K)) ) ).

% atMost_iff
tff(fact_7793_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( member_set_int(I,set_or58775011639299419et_int(K))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(I),K)) ) ).

% atMost_iff
tff(fact_7794_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_ord_atMost_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),K)) ) ).

% atMost_iff
tff(fact_7795_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( pp(aa_set_num_bool(member_num(I),set_ord_atMost_num(K)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),K)) ) ).

% atMost_iff
tff(fact_7796_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).

% atMost_iff
tff(fact_7797_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_atMost_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K)) ) ).

% atMost_iff
tff(fact_7798_cot__zero,axiom,
    aa_complex_complex(cot_complex,zero_zero_complex) = zero_zero_complex ).

% cot_zero
tff(fact_7799_cot__zero,axiom,
    aa_real_real(cot_real,zero_zero_real) = zero_zero_real ).

% cot_zero
tff(fact_7800_finite__atMost,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K))) ).

% finite_atMost
tff(fact_7801_summable__zero,axiom,
    summable_complex(aTP_Lamm_oi) ).

% summable_zero
tff(fact_7802_summable__zero,axiom,
    summable_real(aTP_Lamm_fi) ).

% summable_zero
tff(fact_7803_summable__zero,axiom,
    summable_nat(aTP_Lamm_cb) ).

% summable_zero
tff(fact_7804_summable__zero,axiom,
    summable_int(aTP_Lamm_oj) ).

% summable_zero
tff(fact_7805_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ga(I),F)) ).

% summable_single
tff(fact_7806_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_pr(I),F)) ).

% summable_single
tff(fact_7807_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ps(I),F)) ).

% summable_single
tff(fact_7808_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_pt(I),F)) ).

% summable_single
tff(fact_7809_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ko(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7810_atMost__subset__iff,axiom,
    ! [X2: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X2),set_or58775011639299419et_int(Y))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X2),Y)) ) ).

% atMost_subset_iff
tff(fact_7811_atMost__subset__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(X2)),set_ord_atMost_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y)) ) ).

% atMost_subset_iff
tff(fact_7812_atMost__subset__iff,axiom,
    ! [X2: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(X2)),set_ord_atMost_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Y)) ) ).

% atMost_subset_iff
tff(fact_7813_atMost__subset__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,X2)),aa_nat_set_nat(set_ord_atMost_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ).

% atMost_subset_iff
tff(fact_7814_atMost__subset__iff,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X2)),set_ord_atMost_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y)) ) ).

% atMost_subset_iff
tff(fact_7815_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qx(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7816_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qy(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7817_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1707563613775114915nt_nat(F,A2) = one_one_nat )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
           => ( aa_int_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7818_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups861055069439313189ex_nat(F,A2) = one_one_nat )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
           => ( aa_complex_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7819_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A2) = one_one_nat )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
           => ( aa_nat_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7820_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tv(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7821_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_tw(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7822_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rp(P),F)) ) ).

% summable_If_finite
tff(fact_7823_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_rs(P),F)) ) ).

% summable_If_finite
tff(fact_7824_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rr(P),F)) ) ).

% summable_If_finite
tff(fact_7825_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_rq(P),F)) ) ).

% summable_If_finite
tff(fact_7826_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rt(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7827_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_rw(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7828_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rv(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7829_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ru(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7830_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_7831_and__nat__numerals_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_7832_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_7833_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_7834_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_7835_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_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_7836_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_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_7837_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_7838_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_7839_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : 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,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_7840_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_7841_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_7842_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_7843_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_7844_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : 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,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_7845_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_7846_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A2)))
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7847_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A2)))
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7848_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)))
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7849_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_7850_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_7851_and__nat__numerals_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_7852_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(power_power_real(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_7853_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_7854_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_7855_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_7856_int__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_lw(F),A2) ).

% int_prod
tff(fact_7857_int__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_ly(F),A2) ).

% int_prod
tff(fact_7858_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_7859_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_7860_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_7861_infinite__Iic,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_atMost_int(A))) ).

% infinite_Iic
tff(fact_7862_not__Iic__eq__Icc,axiom,
    ! [H2: int,L: int,H: int] : set_ord_atMost_int(H2) != set_or1266510415728281911st_int(L,H) ).

% not_Iic_eq_Icc
tff(fact_7863_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_7864_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_tx(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7865_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aTP_Lamm_ty(C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7866_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_7867_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_7868_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_7869_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_7870_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(F),G)) ) ) ).

% summable_add
tff(fact_7871_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hi(F),G)) ) ) ).

% summable_add
tff(fact_7872_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_pu(F),G)) ) ) ).

% summable_add
tff(fact_7873_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_ko(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7874_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B2: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N2))),B2))
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_7875_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B2: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B2))
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_7876_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B2: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B2))
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_7877_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_tz,U)) ).

% atMost_def
tff(fact_7878_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_at,U)) ).

% atMost_def
tff(fact_7879_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_av(U)) ).

% atMost_def
tff(fact_7880_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_ua,U)) ).

% atMost_def
tff(fact_7881_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_ub,U)) ).

% atMost_def
tff(fact_7882_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_az,U)) ).

% atMost_def
tff(fact_7883_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_uc,U)) ).

% atMost_def
tff(fact_7884_powser__insidea,axiom,
    ! [F: fun_nat_real,X2: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X2)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ud(F),Z)) ) ) ).

% powser_insidea
tff(fact_7885_powser__insidea,axiom,
    ! [F: fun_nat_complex,X2: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X2)))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_ue(F),Z)) ) ) ).

% powser_insidea
tff(fact_7886_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
     => ( summable_real(F)
       => ( summable_real(G)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),suminf_real(G))) ) ) ) ).

% suminf_le
tff(fact_7887_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G))) ) ) ) ).

% suminf_le
tff(fact_7888_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
     => ( summable_int(F)
       => ( summable_int(G)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),suminf_int(G))) ) ) ) ).

% suminf_le
tff(fact_7889_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_7890_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_7891_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_7892_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_7893_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_or1269000886237332187st_nat(zero_zero_nat,N) ).

% atMost_atLeast0
tff(fact_7894_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_7895_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qx(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7896_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qy(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7897_summable__zero__power,axiom,
    summable_real(power_power_real(zero_zero_real)) ).

% summable_zero_power
tff(fact_7898_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7899_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7900_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_7901_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(F),G)) ) ) ) ).

% suminf_add
tff(fact_7902_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hi(F),G)) ) ) ) ).

% suminf_add
tff(fact_7903_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_pu(F),G)) ) ) ) ).

% suminf_add
tff(fact_7904_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)),set_or1266510415728281911st_int(L2,H2))) ).

% not_Iic_le_Icc
tff(fact_7905_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_7906_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uf(N)))) = groups705719431365010083at_int(aTP_Lamm_uh(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_7907_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uf(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_uj(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.triangle_reindex_eq
tff(fact_7908_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ? [K2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_atMost_nat,K2))) ) ).

% finite_nat_iff_bounded_le
tff(fact_7909_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7910_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7911_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7912_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_nonneg
tff(fact_7913_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_nonneg
tff(fact_7914_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_nonneg
tff(fact_7915_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_pos
tff(fact_7916_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_pos
tff(fact_7917_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_pos
tff(fact_7918_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gn(F)) ).

% summable_zero_power'
tff(fact_7919_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gp(F)) ).

% summable_zero_power'
tff(fact_7920_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_uk(F)) ).

% summable_zero_power'
tff(fact_7921_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gn(F)) ).

% summable_0_powser
tff(fact_7922_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gp(F)) ).

% summable_0_powser
tff(fact_7923_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,N: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ul(N)))) = groups705719431365010083at_int(aTP_Lamm_uh(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_7924_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ul(N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_uj(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.triangle_reindex
tff(fact_7925_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_um(F),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7926_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,M: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_un(F),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7927_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_uo(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7928_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_7929_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_7930_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_7931_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_7932_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_7933_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_ch,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_7934_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).

% suminf_pos2
tff(fact_7935_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).

% suminf_pos2
tff(fact_7936_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).

% suminf_pos2
tff(fact_7937_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
        <=> ? [I5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I5))) ) ) ) ).

% suminf_pos_iff
tff(fact_7938_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
        <=> ? [I5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I5))) ) ) ) ).

% suminf_pos_iff
tff(fact_7939_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
        <=> ? [I5: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I5))) ) ) ) ).

% suminf_pos_iff
tff(fact_7940_suminf__le__const,axiom,
    ! [F: fun_nat_int,X2: int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),X2)) ) ) ).

% suminf_le_const
tff(fact_7941_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X2: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : 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,N2))),X2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),X2)) ) ) ).

% suminf_le_const
tff(fact_7942_suminf__le__const,axiom,
    ! [F: fun_nat_real,X2: real] :
      ( summable_real(F)
     => ( ! [N2: nat] : 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,N2))),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),X2)) ) ) ).

% suminf_le_const
tff(fact_7943_powser__inside,axiom,
    ! [F: fun_nat_real,X2: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X2)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),Z)) ) ) ).

% powser_inside
tff(fact_7944_powser__inside,axiom,
    ! [F: fun_nat_complex,X2: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X2)))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),Z)) ) ) ).

% powser_inside
tff(fact_7945_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X2: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X2))
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7946_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X2: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
     => ( ! [N2: nat] : 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,N2))),X2))
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7947_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X2: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
     => ( ! [N2: nat] : 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,N2))),X2))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7948_complete__algebra__summable__geometric,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),one_one_real))
     => summable_real(power_power_real(X2)) ) ).

% complete_algebra_summable_geometric
tff(fact_7949_complete__algebra__summable__geometric,axiom,
    ! [X2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),one_one_real))
     => summable_complex(power_power_complex(X2)) ) ).

% complete_algebra_summable_geometric
tff(fact_7950_summable__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => summable_real(power_power_real(C)) ) ).

% summable_geometric
tff(fact_7951_summable__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_7952_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_km(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7953_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_lf(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7954_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : 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)),groups3539618377306564664at_int(aTP_Lamm_lg(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7955_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_kl(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7956_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_km(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_7957_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_ni(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_7958_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_nj(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_7959_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nk(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_7960_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,D: fun_nat_complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(D),X3),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
         => ( aa_nat_complex(C,I5) = aa_nat_complex(D,I5) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_7961_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,D: fun_nat_real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(D),X3)),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
         => ( aa_nat_real(C,I5) = aa_nat_real(D,I5) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_7962_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_km(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7963_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_lf(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7964_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : 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)),groups705719431365010083at_int(aTP_Lamm_lg(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7965_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_kl(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_7966_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ch,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_7967_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_up(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ur(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_7968_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_us(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_uu(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_7969_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : groups705719431365010083at_int(aTP_Lamm_uv(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ux(A),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_7970_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_uy(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uz(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_7971_binomial__maximum_H,axiom,
    ! [N: nat,K: 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)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_7972_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_mono
tff(fact_7973_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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)))),K))
       => ( 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),K))) ) ) ) ).

% binomial_antimono
tff(fact_7974_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),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_7975_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I6: set_nat] :
      ( summable_int(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,I6)),suminf_int(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7976_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I6: set_nat] :
      ( summable_nat(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I6)),suminf_nat(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7977_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I6: set_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I6))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I6)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),I6)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7978_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ko(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_7979_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ko(F),K)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_7980_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,K: nat] :
      ( ! [W: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),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,K),N))
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_7981_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,K: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),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,K),N))
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_7982_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
         => ( aa_nat_complex(C,I5) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_7983_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
    <=> ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
         => ( aa_nat_real(C,I5) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_7984_ln__prod,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I6))
     => ( ! [I3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups2703838992350267259T_real(F,I6)) = groups2240296850493347238T_real(aTP_Lamm_va(F),I6) ) ) ) ).

% ln_prod
tff(fact_7985_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_vb(F),I6) ) ) ) ).

% ln_prod
tff(fact_7986_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_vc(F),I6) ) ) ) ).

% ln_prod
tff(fact_7987_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_vd(F),I6) ) ) ) ).

% ln_prod
tff(fact_7988_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_ve(F),I6) ) ) ) ).

% ln_prod
tff(fact_7989_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_vf(F)),I6) ) ) ) ).

% ln_prod
tff(fact_7990_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_lf(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7991_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : 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)),groups3539618377306564664at_int(aTP_Lamm_lg(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7992_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_kl(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7993_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_km(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_7994_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,M: 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,M),N))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_7995_sum__up__index__split,axiom,
    ! [F: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_7996_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,M: 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,M),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,M))),aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_7997_sum__up__index__split,axiom,
    ! [F: fun_nat_real,M: 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,M),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,M))),aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))) ).

% sum_up_index_split
tff(fact_7998_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_km(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_7999_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_lf(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8000_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : 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)),groups705719431365010083at_int(aTP_Lamm_lg(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8001_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_kl(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8002_atLeast1__atMost__eq__remove0,axiom,
    ! [N: 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_8003_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uf(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_vg(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.triangle_reindex_eq
tff(fact_8004_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uf(N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vi(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.triangle_reindex_eq
tff(fact_8005_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M3))) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ).

% sum_less_suminf
tff(fact_8006_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ).

% sum_less_suminf
tff(fact_8007_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M3))) )
       => 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,N))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_8008_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),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),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_8009_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),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),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_strict_mono
tff(fact_8010_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),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))),K)))
       => ( 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),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_8011_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vj(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8012_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_vk(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8013_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vj(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8014_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_vk(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ts(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8015_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,set_or1269000886237332187st_nat(M2,N5)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8016_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(M2,N5)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8017_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ko(F),N5)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_8018_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_vl(F),N5)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_8019_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_ts(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8020_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),R0))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N2))),aa_nat_real(power_power_real(R0),N2))),M5))
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_vm(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_8021_sum__gp__basic,axiom,
    ! [X2: complex,N: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X2)),groups2073611262835488442omplex(power_power_complex(X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8022_sum__gp__basic,axiom,
    ! [X2: rat,N: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X2)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8023_sum__gp__basic,axiom,
    ! [X2: 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),X2)),groups3539618377306564664at_int(power_power_int(X2),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(X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8024_sum__gp__basic,axiom,
    ! [X2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X2),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8025_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_vn(C),N))))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
          & ( aa_nat_complex(C,I5) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_8026_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_vo(C),N))))
    <=> ? [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
          & ( aa_nat_real(C,I5) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_8027_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,N: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_vn(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_8028_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,N: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_vo(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_8029_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,N: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),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_tj(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8030_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tm(C),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_tm(C),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_tm(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8031_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,N: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(C),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(C),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)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8032_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,N: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),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_ts(C),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_ts(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8033_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,N: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),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_tj(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8034_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,N: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tm(C),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_tm(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tm(C),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8035_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,N: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(C),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)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(C),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8036_polyfun__linear__factor,axiom,
    ! [C: 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_ts(C),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_ts(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8037_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2073611262835488442omplex(power_power_complex(X2),set_or1269000886237332187st_nat(M,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X2),M)),groups2073611262835488442omplex(power_power_complex(X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8038_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(power_power_rat(X2),set_or1269000886237332187st_nat(M,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X2),M)),groups2906978787729119204at_rat(power_power_rat(X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8039_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(power_power_int(X2),set_or1269000886237332187st_nat(M,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X2),M)),groups3539618377306564664at_int(power_power_int(X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8040_sum__power__shift,axiom,
    ! [M: nat,N: nat,X2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),set_or1269000886237332187st_nat(M,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X2),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8041_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8042_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8043_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8044_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8045_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
           => 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,N))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8046_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,N: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ul(N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_vg(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.triangle_reindex
tff(fact_8047_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,N: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ul(N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vi(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.triangle_reindex
tff(fact_8048_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_lr(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8049_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_ls(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8050_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_lt(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8051_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_lu(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8052_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_complex,N: nat,B: fun_nat_complex,X2: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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_tj(A),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(B),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_vq(A),B),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8053_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_rat,N: nat,B: fun_nat_rat,X2: rat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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_tm(A),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tm(B),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_vs(A),B),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8054_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_int,N: nat,B: fun_nat_int,X2: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(A),X2),aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(B),X2),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_vu(A),B),X2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8055_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_real,N: nat,B: fun_nat_real,X2: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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_ts(A),X2)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(B),X2)),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_vw(A),B),X2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product
tff(fact_8056_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_ta(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8057_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_tb(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8058_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_tc(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8059_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_td(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8060_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,N: nat,K: complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),X3),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,N)))
           => ( aa_nat_complex(C,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8061_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,N: nat,K: real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,N)))
           => ( aa_nat_real(C,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8062_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K3: int,L3: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K3,L3))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K3),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(P,aa_int_int(divide_divide_int(K3),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(P,K3),L3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_8063_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X2: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),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_vx(A),X2)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vx(B),X2)),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_vz(A),B),X2)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_8064_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),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,M),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,M),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_8065_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),M))),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,M),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_8066_sum_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wa(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wb(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8067_sum_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wc(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wd(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8068_sum_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_we(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wf(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8069_sum_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wg(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wh(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8070_sum_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wi(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wj(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8071_prod_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wk(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wb(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8072_prod_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wl(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wj(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8073_prod_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wm(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wd(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8074_prod_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wn(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wf(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8075_prod_Ozero__middle,axiom,
    ! [P3: nat,K: 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),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wo(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wh(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8076_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_wp(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_8077_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 )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wq(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_8078_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_wr(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_8079_root__polyfun,axiom,
    ! [N: nat,Z: code_integer,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),N) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_ws(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_8080_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(power_power_real(Z),N) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wt(N),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_8081_sum__gp0,axiom,
    ! [X2: complex,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X2),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)) ) )
      & ( ( X2 != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X2),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(X2),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X2)) ) ) ) ).

% sum_gp0
tff(fact_8082_sum__gp0,axiom,
    ! [X2: rat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),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)) ) )
      & ( ( X2 != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X2),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(X2),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X2)) ) ) ) ).

% sum_gp0
tff(fact_8083_sum__gp0,axiom,
    ! [X2: real,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),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)) ) )
      & ( ( X2 != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X2)),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(power_power_real(X2),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X2)) ) ) ) ).

% sum_gp0
tff(fact_8084_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_complex,X2: 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_tj(A),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X2),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_wv(N),A),X2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8085_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_rat,X2: 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_tm(A),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tm(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X2),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_wx(N),A),X2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8086_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_int,X2: 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,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(A),X2),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tp(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,X2),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_wz(N),A),X2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8087_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_real,X2: 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_ts(A),X2)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(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,X2),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_xb(N),A),X2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8088_cot__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X2))) ) ) ).

% cot_gt_zero
tff(fact_8089_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: complex] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_complex_real(real_V1022390504157884413omplex,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z3)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8090_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_real_real(real_V7735802525324610683m_real,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,Z3)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8091_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_xc(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_8092_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_xd(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_8093_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(aTP_Lamm_xe(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8094_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))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xf,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_8095_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_xg,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8096_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_xh(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_8097_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_xi(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_8098_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),groups7501900531339628137nteger(aTP_Lamm_xj(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8099_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))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xk,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_8100_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_xl,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8101_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_8102_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_8103_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups7501900531339628137nteger(aTP_Lamm_xo(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8104_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xp,N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8105_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_xq,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8106_binomial__r__part__sum,axiom,
    ! [M: 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))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_8107_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_xr(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8108_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_xs(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8109_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_xt(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8110_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xu,N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8111_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xv,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8112_zero__less__binomial__iff,axiom,
    ! [N: nat,K: 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),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_8113_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_8114_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_8115_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_8116_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_8117_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_8118_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8119_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_8120_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_8121_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_8122_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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),K))) ) ).

% zero_less_binomial
tff(fact_8123_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_8124_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_8125_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xw,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_8126_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xx,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = 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),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8127_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xx,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = 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),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8128_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_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,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8129_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8130_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),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_8131_choose__reduce__nat,axiom,
    ! [N: nat,K: 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),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8132_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: 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(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = 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,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8133_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xy(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8134_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_xz(M),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,M),N)),R2) ).

% vandermonde
tff(fact_8135_binomial__addition__formula,axiom,
    ! [N: nat,K: 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,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8136_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_ya(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_8137_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_yb(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8138_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_yc(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8139_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) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_yd(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8140_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_ya(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8141_binomial__ring,axiom,
    ! [A: real,B: real,N: nat] : aa_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_ye(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8142_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_yf(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8143_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) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_yg(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8144_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N: nat] : aa_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_yh(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8145_summable__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => summable_real(aTP_Lamm_yi(X2)) ) ).

% summable_arctan_series
tff(fact_8146_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I3: int,J2: int] :
            ( 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(P,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(P,I3),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_8147_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_yj(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8148_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_yk(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8149_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yl(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8150_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ym(C),X2))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_yn(C),X2)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ym(C),X2)))) ) ).

% diffs_equiv
tff(fact_8151_diffs__equiv,axiom,
    ! [C: fun_nat_real,X2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_yo(C),X2))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_yp(C),X2)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_yo(C),X2)))) ) ).

% diffs_equiv
tff(fact_8152_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,M))),one_one_complex)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8153_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(gbinomial_rat(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,M))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8154_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(gbinomial_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))),aa_nat_real(semiri5074537144036343181t_real,M))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8155_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_8156_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_8157_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_8158_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_8159_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_8160_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_8161_abs__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_abs
tff(fact_8162_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_8163_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_8164_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_8165_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_8166_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_8167_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_8168_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_8169_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_8170_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_8171_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_8172_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_8173_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_8174_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_8175_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_8176_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_8177_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_8178_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_8179_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_8180_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_8181_abs__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_numeral
tff(fact_8182_abs__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).

% abs_numeral
tff(fact_8183_abs__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,N) ).

% abs_numeral
tff(fact_8184_abs__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(N) ).

% abs_numeral
tff(fact_8185_abs__mult__self__eq,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(A)) = times_3573771949741848930nteger(A,A) ).

% abs_mult_self_eq
tff(fact_8186_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_8187_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_8188_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_8189_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_8190_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_8191_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_8192_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_8193_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_8194_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_8195_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_8196_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_8197_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_8198_abs__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_8199_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_8200_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_8201_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_8202_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_8203_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_8204_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_8205_abs__minus,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus
tff(fact_8206_abs__minus,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus
tff(fact_8207_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_8208_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% abs_dvd_iff
tff(fact_8209_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% abs_dvd_iff
tff(fact_8210_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,M)),K))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% abs_dvd_iff
tff(fact_8211_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% abs_dvd_iff
tff(fact_8212_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),aa_real_real(abs_abs_real,K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% dvd_abs_iff
tff(fact_8213_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),aa_int_int(abs_abs_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% dvd_abs_iff
tff(fact_8214_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),aa_rat_rat(abs_abs_rat,K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% dvd_abs_iff
tff(fact_8215_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% dvd_abs_iff
tff(fact_8216_abs__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_8217_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_8218_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_8219_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_8220_of__int__abs,axiom,
    ! [X2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,X2)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,X2)) ).

% of_int_abs
tff(fact_8221_of__int__abs,axiom,
    ! [X2: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,X2)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,X2)) ).

% of_int_abs
tff(fact_8222_of__int__abs,axiom,
    ! [X2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,X2)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,X2)) ).

% of_int_abs
tff(fact_8223_of__int__abs,axiom,
    ! [X2: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X2)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,X2)) ).

% of_int_abs
tff(fact_8224_abs__bool__eq,axiom,
    ! [P: bool] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real(P)) = zero_n3304061248610475627l_real(P) ).

% abs_bool_eq
tff(fact_8225_abs__bool__eq,axiom,
    ! [P: bool] : aa_rat_rat(abs_abs_rat,zero_n2052037380579107095ol_rat(P)) = zero_n2052037380579107095ol_rat(P) ).

% abs_bool_eq
tff(fact_8226_abs__bool__eq,axiom,
    ! [P: bool] : abs_abs_Code_integer(zero_n356916108424825756nteger(P)) = zero_n356916108424825756nteger(P) ).

% abs_bool_eq
tff(fact_8227_abs__bool__eq,axiom,
    ! [P: bool] : aa_int_int(abs_abs_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% abs_bool_eq
tff(fact_8228_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_8229_abs__of__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_8230_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_8231_abs__of__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_8232_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% abs_le_self_iff
tff(fact_8233_abs__le__self__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% abs_le_self_iff
tff(fact_8234_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% abs_le_self_iff
tff(fact_8235_abs__le__self__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% abs_le_self_iff
tff(fact_8236_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_8237_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_8238_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_8239_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_8240_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_8241_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_8242_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_8243_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_8244_abs__neg__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).

% abs_neg_numeral
tff(fact_8245_abs__neg__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,N) ).

% abs_neg_numeral
tff(fact_8246_abs__neg__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_neg_numeral
tff(fact_8247_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(N) ).

% abs_neg_numeral
tff(fact_8248_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_8249_abs__neg__one,axiom,
    aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).

% abs_neg_one
tff(fact_8250_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_8251_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_8252_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_8253_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(gbinomial_real(zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_8254_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_8255_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_8256_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_8257_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_8258_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(gbinomial_real(A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_8259_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_8260_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_8261_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_8262_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_yq(F),A2))) ).

% sum_abs
tff(fact_8263_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yr(F)),A2))) ).

% sum_abs
tff(fact_8264_zero__le__divide__abs__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),aa_real_real(abs_abs_real,B))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_8265_zero__le__divide__abs__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),aa_rat_rat(abs_abs_rat,B))))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_8266_divide__le__0__abs__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),aa_real_real(abs_abs_real,B))),zero_zero_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_8267_divide__le__0__abs__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),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_8268_abs__of__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_8269_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_8270_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_8271_abs__of__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_8272_artanh__minus__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X2)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X2)) ) ) ).

% artanh_minus_real
tff(fact_8273_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_yq(F),A2))) ).

% sum_abs_ge_zero
tff(fact_8274_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yr(F)),A2))) ).

% sum_abs_ge_zero
tff(fact_8275_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N)))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8276_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)))
    <=> ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8277_zero__less__power__abs__iff,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(aa_rat_rat(abs_abs_rat,A)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8278_zero__less__power__abs__iff,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(aa_int_int(abs_abs_int,A)),N)))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8279_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% abs_le_D1
tff(fact_8280_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% abs_le_D1
tff(fact_8281_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% abs_le_D1
tff(fact_8282_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% abs_le_D1
tff(fact_8283_abs__ge__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).

% abs_ge_self
tff(fact_8284_abs__ge__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A))) ).

% abs_ge_self
tff(fact_8285_abs__ge__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_self
tff(fact_8286_abs__ge__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).

% abs_ge_self
tff(fact_8287_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_8288_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_8289_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_8290_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_8291_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_8292_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(times_3573771949741848930nteger(A,B)) = times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_8293_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_8294_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_8295_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_8296_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_8297_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_8298_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_8299_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_8300_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_8301_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_8302_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_8303_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_8304_abs__eq__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X2) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X2 = Y )
        | ( X2 = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_8305_abs__eq__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X2) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X2 = Y )
        | ( X2 = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_8306_abs__eq__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,X2) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( X2 = Y )
        | ( X2 = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_8307_abs__eq__iff,axiom,
    ! [X2: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(X2) = abs_abs_Code_integer(Y) )
    <=> ( ( X2 = Y )
        | ( X2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_8308_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_8309_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_8310_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_8311_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_8312_abs__ge__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))) ).

% abs_ge_zero
tff(fact_8313_abs__ge__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).

% abs_ge_zero
tff(fact_8314_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_zero
tff(fact_8315_abs__ge__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).

% abs_ge_zero
tff(fact_8316_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_8317_abs__of__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_8318_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_8319_abs__of__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_8320_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)) ).

% abs_not_less_zero
tff(fact_8321_abs__not__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_8322_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_8323_abs__not__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_8324_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq
tff(fact_8325_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq
tff(fact_8326_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq
tff(fact_8327_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : 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(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq
tff(fact_8328_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),times_3573771949741848930nteger(C,D))) ) ) ).

% abs_mult_less
tff(fact_8329_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).

% abs_mult_less
tff(fact_8330_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D))) ) ) ).

% abs_mult_less
tff(fact_8331_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B)),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(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))) ) ) ).

% abs_mult_less
tff(fact_8332_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)))) ).

% abs_triangle_ineq2_sym
tff(fact_8333_abs__triangle__ineq2__sym,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(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_8334_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_8335_abs__triangle__ineq2__sym,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(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_8336_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq3
tff(fact_8337_abs__triangle__ineq3,axiom,
    ! [A: real,B: 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,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq3
tff(fact_8338_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq3
tff(fact_8339_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : 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_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq3
tff(fact_8340_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq2
tff(fact_8341_abs__triangle__ineq2,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(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq2
tff(fact_8342_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq2
tff(fact_8343_abs__triangle__ineq2,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(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq2
tff(fact_8344_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_8345_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_8346_abs__ge__minus__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).

% abs_ge_minus_self
tff(fact_8347_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A))) ).

% abs_ge_minus_self
tff(fact_8348_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_minus_self
tff(fact_8349_abs__ge__minus__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).

% abs_ge_minus_self
tff(fact_8350_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_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,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_le_iff
tff(fact_8351_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_le_iff
tff(fact_8352_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_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,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_le_iff
tff(fact_8353_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_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,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_le_iff
tff(fact_8354_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ).

% abs_le_D2
tff(fact_8355_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).

% abs_le_D2
tff(fact_8356_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ).

% abs_le_D2
tff(fact_8357_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).

% abs_le_D2
tff(fact_8358_abs__leI,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,A)),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)) ) ) ).

% abs_leI
tff(fact_8359_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).

% abs_leI
tff(fact_8360_abs__leI,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,A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)) ) ) ).

% abs_leI
tff(fact_8361_abs__leI,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,A)),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).

% abs_leI
tff(fact_8362_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( 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,A)),B)) ) ) ).

% abs_less_iff
tff(fact_8363_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( 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,A)),B)) ) ) ).

% abs_less_iff
tff(fact_8364_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( 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,A)),B)) ) ) ).

% abs_less_iff
tff(fact_8365_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_less_iff
tff(fact_8366_dense__eq0__I,axiom,
    ! [X2: 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,aa_real_real(abs_abs_real,X2)),E)) )
     => ( X2 = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_8367_dense__eq0__I,axiom,
    ! [X2: 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,aa_rat_rat(abs_abs_rat,X2)),E)) )
     => ( X2 = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_8368_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
     => ( abs_abs_Code_integer(times_3573771949741848930nteger(A,B)) = times_3573771949741848930nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_8369_abs__eq__mult,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,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,B),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_8370_abs__eq__mult,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,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,B),zero_zero_rat)) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_8371_abs__eq__mult,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,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,B),zero_zero_int)) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_8372_abs__mult__pos,axiom,
    ! [X2: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2))
     => ( times_3573771949741848930nteger(abs_abs_Code_integer(Y),X2) = abs_abs_Code_integer(times_3573771949741848930nteger(Y,X2)) ) ) ).

% abs_mult_pos
tff(fact_8373_abs__mult__pos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X2) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X2)) ) ) ).

% abs_mult_pos
tff(fact_8374_abs__mult__pos,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y)),X2) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),X2)) ) ) ).

% abs_mult_pos
tff(fact_8375_abs__mult__pos,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X2) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X2)) ) ) ).

% abs_mult_pos
tff(fact_8376_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8377_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8378_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8379_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8380_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8381_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8382_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8383_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8384_abs__minus__le__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real)) ).

% abs_minus_le_zero
tff(fact_8385_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_8386_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_8387_abs__minus__le__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).

% abs_minus_le_zero
tff(fact_8388_abs__div__pos,axiom,
    ! [Y: rat,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X2)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X2),Y)) ) ) ).

% abs_div_pos
tff(fact_8389_abs__div__pos,axiom,
    ! [Y: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,X2)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),Y)) ) ) ).

% abs_div_pos
tff(fact_8390_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N))) ).

% zero_le_power_abs
tff(fact_8391_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N))) ).

% zero_le_power_abs
tff(fact_8392_zero__le__power__abs,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(aa_rat_rat(abs_abs_rat,A)),N))) ).

% zero_le_power_abs
tff(fact_8393_zero__le__power__abs,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(aa_int_int(abs_abs_int,A)),N))) ).

% zero_le_power_abs
tff(fact_8394_abs__if__raw,axiom,
    ! [X: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X) = aa_int_int(uminus_uminus_int,X) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X) = X ) ) ) ).

% abs_if_raw
tff(fact_8395_abs__if__raw,axiom,
    ! [X: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X) = aa_real_real(uminus_uminus_real,X) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X) = X ) ) ) ).

% abs_if_raw
tff(fact_8396_abs__if__raw,axiom,
    ! [X: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X) = aa_rat_rat(uminus_uminus_rat,X) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X) = X ) ) ) ).

% abs_if_raw
tff(fact_8397_abs__if__raw,axiom,
    ! [X: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X) = X ) ) ) ).

% abs_if_raw
tff(fact_8398_abs__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_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))
       => ( aa_int_int(abs_abs_int,A) = A ) ) ) ).

% abs_if
tff(fact_8399_abs__if,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_if
tff(fact_8400_abs__if,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_if
tff(fact_8401_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = A ) ) ) ).

% abs_if
tff(fact_8402_abs__of__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_8403_abs__of__neg,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) ) ) ).

% abs_of_neg
tff(fact_8404_abs__of__neg,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) ) ) ).

% abs_of_neg
tff(fact_8405_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_8406_abs__diff__le__iff,axiom,
    ! [X2: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X2,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),X2))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X2),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_le_iff
tff(fact_8407_abs__diff__le__iff,axiom,
    ! [X2: real,A: real,R2: 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,X2),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_8408_abs__diff__le__iff,axiom,
    ! [X2: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X2),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),X2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),aa_rat_rat(plus_plus_rat(A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_8409_abs__diff__le__iff,axiom,
    ! [X2: int,A: int,R2: int] :
      ( 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,X2),A))),R2))
    <=> ( 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),R2)),X2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_8410_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D))))) ).

% abs_diff_triangle_ineq
tff(fact_8411_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: 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,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_8412_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_8413_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : 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_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_8414_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq4
tff(fact_8415_abs__triangle__ineq4,axiom,
    ! [A: real,B: 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,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq4
tff(fact_8416_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq4
tff(fact_8417_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : 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,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq4
tff(fact_8418_abs__diff__less__iff,axiom,
    ! [X2: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X2,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),X2))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_less_iff
tff(fact_8419_abs__diff__less__iff,axiom,
    ! [X2: real,A: real,R2: 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,X2),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_8420_abs__diff__less__iff,axiom,
    ! [X2: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X2),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),X2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),aa_rat_rat(plus_plus_rat(A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_8421_abs__diff__less__iff,axiom,
    ! [X2: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_8422_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8423_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8424_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8425_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_8426_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),K) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8427_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_yr(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8428_abs__add__one__gt__zero,axiom,
    ! [X2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X2)))) ).

% abs_add_one_gt_zero
tff(fact_8429_abs__add__one__gt__zero,axiom,
    ! [X2: real] : 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),aa_real_real(abs_abs_real,X2)))) ).

% abs_add_one_gt_zero
tff(fact_8430_abs__add__one__gt__zero,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,X2)))) ).

% abs_add_one_gt_zero
tff(fact_8431_abs__add__one__gt__zero,axiom,
    ! [X2: 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,one_one_int),aa_int_int(abs_abs_int,X2)))) ).

% abs_add_one_gt_zero
tff(fact_8432_of__int__leD,axiom,
    ! [N: int,X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),X2)) ) ) ).

% of_int_leD
tff(fact_8433_of__int__leD,axiom,
    ! [N: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ) ).

% of_int_leD
tff(fact_8434_of__int__leD,axiom,
    ! [N: int,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X2)) ) ) ).

% of_int_leD
tff(fact_8435_of__int__leD,axiom,
    ! [N: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X2)) ) ) ).

% of_int_leD
tff(fact_8436_of__int__lessD,axiom,
    ! [N: int,X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X2)) ) ) ).

% of_int_lessD
tff(fact_8437_of__int__lessD,axiom,
    ! [N: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ) ).

% of_int_lessD
tff(fact_8438_of__int__lessD,axiom,
    ! [N: int,X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X2)) ) ) ).

% of_int_lessD
tff(fact_8439_of__int__lessD,axiom,
    ! [N: int,X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X2))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X2)) ) ) ).

% of_int_lessD
tff(fact_8440_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_8441_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_8442_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_8443_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(gbinomial_real(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8444_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A))
     => 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(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8445_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8446_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8447_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8448_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8449_round__diff__minimal,axiom,
    ! [Z: real,M: int] : 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,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,M))))) ).

% round_diff_minimal
tff(fact_8450_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,M))))) ).

% round_diff_minimal
tff(fact_8451_abs__le__square__iff,axiom,
    ! [X2: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X2)),abs_abs_Code_integer(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_8452_abs__le__square__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),aa_real_real(abs_abs_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_8453_abs__le__square__iff,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X2)),aa_rat_rat(abs_abs_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_8454_abs__le__square__iff,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X2)),aa_int_int(abs_abs_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_8455_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_8456_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_8457_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(gbinomial_real(A),K)) ).

% Suc_times_gbinomial
tff(fact_8458_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8459_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),M)),aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8460_gbinomial__parallel__sum,axiom,
    ! [A: complex,N: nat] : groups2073611262835488442omplex(aTP_Lamm_ys(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),N) ).

% gbinomial_parallel_sum
tff(fact_8461_gbinomial__parallel__sum,axiom,
    ! [A: rat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_yt(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),N) ).

% gbinomial_parallel_sum
tff(fact_8462_gbinomial__parallel__sum,axiom,
    ! [A: real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yu(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),N) ).

% gbinomial_parallel_sum
tff(fact_8463_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X2)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_8464_power2__le__iff__abs__le,axiom,
    ! [Y: real,X2: 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_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_8465_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X2: 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_nat_rat(power_power_rat(X2),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,aa_rat_rat(abs_abs_rat,X2)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_8466_power2__le__iff__abs__le,axiom,
    ! [Y: int,X2: 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_eq_int,aa_nat_int(power_power_int(X2),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,aa_int_int(abs_abs_int,X2)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_8467_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X2: code_integer] :
      ( ! [X4: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X4),aa_nat_Code_integer(power_8256067586552552935nteger(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,abs_abs_Code_integer(X2)),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_8468_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X2: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X4),aa_nat_real(power_power_real(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X2)),aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_8469_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X2: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X4),aa_nat_rat(power_power_rat(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,aa_rat_rat(abs_abs_rat,X2)),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_8470_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X2: int] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X4),aa_nat_int(power_power_int(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X2)),aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_8471_abs__square__le__1,axiom,
    ! [X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X2)),one_one_Code_integer)) ) ).

% abs_square_le_1
tff(fact_8472_abs__square__le__1,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real)) ) ).

% abs_square_le_1
tff(fact_8473_abs__square__le__1,axiom,
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X2)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_8474_abs__square__le__1,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X2)),one_one_int)) ) ).

% abs_square_le_1
tff(fact_8475_abs__square__less__1,axiom,
    ! [X2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X2)),one_one_Code_integer)) ) ).

% abs_square_less_1
tff(fact_8476_abs__square__less__1,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real)) ) ).

% abs_square_less_1
tff(fact_8477_abs__square__less__1,axiom,
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X2)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_8478_abs__square__less__1,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X2)),one_one_int)) ) ).

% abs_square_less_1
tff(fact_8479_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( 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_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).

% power_mono_even
tff(fact_8480_power__mono__even,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,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ).

% power_mono_even
tff(fact_8481_power__mono__even,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,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,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_even
tff(fact_8482_power__mono__even,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,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,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_even
tff(fact_8483_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8484_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8485_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8486_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_8487_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_8488_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(gbinomial_real(A),K)) ).

% gbinomial_factors
tff(fact_8489_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X2: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I6))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X2,I3))) )
     => ( ( groups7501900531339628137nteger(X2,I6) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I6))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_yv(X2),A),I6),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8490_convex__sum__bound__le,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VE9219535225822561060nteger,A: fun_VE9219535225822561060nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(X2,I3))) )
     => ( ( groups5748017345553531991nteger(X2,I6) = one_one_Code_integer )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_VEB5935388676523804373nteger(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups5748017345553531991nteger(aa_fun2400605388621188028nteger(aTP_Lamm_yw(X2),A),I6),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8491_convex__sum__bound__le,axiom,
    ! [I6: set_int,X2: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I6))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X2,I3))) )
     => ( ( groups7873554091576472773nteger(X2,I6) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I6))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_yx(X2),A),I6),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8492_convex__sum__bound__le,axiom,
    ! [I6: set_real,X2: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I6))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X2,I3))) )
     => ( ( groups7713935264441627589nteger(X2,I6) = one_one_Code_integer )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_yy(X2),A),I6),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8493_convex__sum__bound__le,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VEBT_VEBT_real,A: fun_VEBT_VEBT_real,B: real,Delta: real] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(X2,I3))) )
     => ( ( groups2240296850493347238T_real(X2,I6) = one_one_real )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => 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,aa_VEBT_VEBT_real(A,I3)),B))),Delta)) )
         => 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,groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_yz(X2),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8494_convex__sum__bound__le,axiom,
    ! [I6: set_int,X2: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [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(X2,I3))) )
     => ( ( groups8778361861064173332t_real(X2,I6) = one_one_real )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I6))
             => 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,aa_int_real(A,I3)),B))),Delta)) )
         => 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,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_za(X2),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8495_convex__sum__bound__le,axiom,
    ! [I6: set_real,X2: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [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(X2,I3))) )
     => ( ( groups8097168146408367636l_real(X2,I6) = one_one_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I6))
             => 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,aa_real_real(A,I3)),B))),Delta)) )
         => 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,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_zb(X2),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8496_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X2: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [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(X2,I3))) )
     => ( ( groups2906978787729119204at_rat(X2,I6) = one_one_rat )
       => ( ! [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_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_zc(X2),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8497_convex__sum__bound__le,axiom,
    ! [I6: set_VEBT_VEBT,X2: fun_VEBT_VEBT_rat,A: fun_VEBT_VEBT_rat,B: rat,Delta: rat] :
      ( ! [I3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(X2,I3))) )
     => ( ( groups136491112297645522BT_rat(X2,I6) = one_one_rat )
       => ( ! [I3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I3),I6))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_VEBT_VEBT_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_zd(X2),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8498_convex__sum__bound__le,axiom,
    ! [I6: set_int,X2: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [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(X2,I3))) )
     => ( ( groups3906332499630173760nt_rat(X2,I6) = one_one_rat )
       => ( ! [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_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ze(X2),A),I6)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8499_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_8500_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_8501_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_8502_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8503_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8504_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8505_termdiff__converges,axiom,
    ! [X2: real,K5: real,C: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),K5))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X4)),K5))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ts(C),X4)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_zf(X2),C)) ) ) ).

% termdiff_converges
tff(fact_8506_termdiff__converges,axiom,
    ! [X2: complex,K5: real,C: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),K5))
     => ( ! [X4: complex] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X4)),K5))
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(C),X4)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_zg(X2),C)) ) ) ).

% termdiff_converges
tff(fact_8507_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X2: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zh(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zi(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8508_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X2: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zj(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zk(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8509_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X2: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zl(M),A),X2),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zm(M),A),X2),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8510_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2073611262835488442omplex(aTP_Lamm_zn(K),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8511_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_zo(K),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8512_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_zp,K)),set_or1269000886237332187st_nat(zero_zero_nat,N)) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8513_of__int__round__abs__le,axiom,
    ! [X2: 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,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X2))),X2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_8514_of__int__round__abs__le,axiom,
    ! [X2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X2))),X2))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_8515_round__unique_H,axiom,
    ! [X2: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X2),aa_int_rat(ring_1_of_int_rat,N)))),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,X2) = N ) ) ).

% round_unique'
tff(fact_8516_round__unique_H,axiom,
    ! [X2: real,N: int] :
      ( 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,X2),aa_int_real(ring_1_of_int_real,N)))),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,X2) = N ) ) ).

% round_unique'
tff(fact_8517_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8518_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8519_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8520_real__sqrt__sum__squares__less,axiom,
    ! [X2: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8521_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups2073611262835488442omplex(aTP_Lamm_zq(M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8522_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_zr,M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8523_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X2: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zh(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zs(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8524_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X2: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zj(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zt(M),A),X2),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8525_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X2: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zl(M),A),X2),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zu(M),A),X2),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8526_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_yk(R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8527_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yl(R2)),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8528_lemma__interval,axiom,
    ! [A: real,X2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [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,X2),Y3))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),B)) ) ) ) ) ) ).

% lemma_interval
tff(fact_8529_monoseq__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => topolo6980174941875973593q_real(aTP_Lamm_zv(X2)) ) ).

% monoseq_arctan_series
tff(fact_8530_arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => ( aa_real_real(arctan,X2) = suminf_real(aTP_Lamm_yi(X2)) ) ) ).

% arctan_series
tff(fact_8531_lemma__interval__lt,axiom,
    ! [A: real,X2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
       => ? [D4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D4))
            & ! [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,X2),Y3))),D4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_8532_arctan__double,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),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,X2)) = 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))),X2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_8533_zero__less__arctan__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% zero_less_arctan_iff
tff(fact_8534_arctan__less__zero__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ).

% arctan_less_zero_iff
tff(fact_8535_zdvd1__eq,axiom,
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),one_one_int))
    <=> ( aa_int_int(abs_abs_int,X2) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_8536_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_8537_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_8538_arctan__monotone,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X2)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone
tff(fact_8539_arctan__less__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X2)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% arctan_less_iff
tff(fact_8540_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_8541_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M4: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M4),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S2)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_8542_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M4: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M4),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8543_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_8544_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
       => 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,I))) ) ) ).

% dvd_imp_le_int
tff(fact_8545_abs__mod__less,axiom,
    ! [L: int,K: 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(K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_8546_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != 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,M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_8547_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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,M),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
           => ? [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_8548_decr__lemma,axiom,
    ! [D: int,X2: 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,X2),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,X2),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_8549_incr__lemma,axiom,
    ! [D: int,Z: int,X2: 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,X2),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,X2),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_8550_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_8551_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: 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)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),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) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_8552_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_8553_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_8554_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: 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)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),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) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_8555_arctan__add,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),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,X2)),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,X2),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),Y)))) ) ) ) ).

% arctan_add
tff(fact_8556_monoI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,M3)),aa_nat_real(X6,N2))) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI1
tff(fact_8557_monoI1,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,M3)),aa_nat_set_int(X6,N2))) )
     => topolo3100542954746470799et_int(X6) ) ).

% monoI1
tff(fact_8558_monoI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,M3)),aa_nat_rat(X6,N2))) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI1
tff(fact_8559_monoI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,M3)),aa_nat_num(X6,N2))) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI1
tff(fact_8560_monoI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,M3)),aa_nat_nat(X6,N2))) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI1
tff(fact_8561_monoI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,M3)),aa_nat_int(X6,N2))) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI1
tff(fact_8562_monoI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N2)),aa_nat_real(X6,M3))) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI2
tff(fact_8563_monoI2,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N2)),aa_nat_set_int(X6,M3))) )
     => topolo3100542954746470799et_int(X6) ) ).

% monoI2
tff(fact_8564_monoI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N2)),aa_nat_rat(X6,M3))) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI2
tff(fact_8565_monoI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N2)),aa_nat_num(X6,M3))) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI2
tff(fact_8566_monoI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N2)),aa_nat_nat(X6,M3))) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI2
tff(fact_8567_monoI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N2)),aa_nat_int(X6,M3))) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI2
tff(fact_8568_monoseq__def,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,M4)),aa_nat_real(X6,N3))) )
        | ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,M4))) ) ) ) ).

% monoseq_def
tff(fact_8569_monoseq__def,axiom,
    ! [X6: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,M4)),aa_nat_set_int(X6,N3))) )
        | ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N3)),aa_nat_set_int(X6,M4))) ) ) ) ).

% monoseq_def
tff(fact_8570_monoseq__def,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,M4)),aa_nat_rat(X6,N3))) )
        | ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,M4))) ) ) ) ).

% monoseq_def
tff(fact_8571_monoseq__def,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,M4)),aa_nat_num(X6,N3))) )
        | ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,M4))) ) ) ) ).

% monoseq_def
tff(fact_8572_monoseq__def,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,M4)),aa_nat_nat(X6,N3))) )
        | ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,M4))) ) ) ) ).

% monoseq_def
tff(fact_8573_monoseq__def,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,M4)),aa_nat_int(X6,N3))) )
        | ! [M4: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,M4))) ) ) ) ).

% monoseq_def
tff(fact_8574_monoseq__Suc,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N3)),aa_nat_real(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N3))),aa_nat_real(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8575_monoseq__Suc,axiom,
    ! [X6: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X6)
    <=> ( ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N3)),aa_nat_set_int(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,aa_nat_nat(suc,N3))),aa_nat_set_int(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8576_monoseq__Suc,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_nat_rat(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N3))),aa_nat_rat(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8577_monoseq__Suc,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N3)),aa_nat_num(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N3))),aa_nat_num(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8578_monoseq__Suc,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N3)),aa_nat_nat(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N3))),aa_nat_nat(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8579_monoseq__Suc,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N3)),aa_nat_int(X6,aa_nat_nat(suc,N3))))
        | ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N3))),aa_nat_int(X6,N3))) ) ) ).

% monoseq_Suc
tff(fact_8580_mono__SucI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N2))),aa_nat_real(X6,N2)))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI2
tff(fact_8581_mono__SucI2,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,aa_nat_nat(suc,N2))),aa_nat_set_int(X6,N2)))
     => topolo3100542954746470799et_int(X6) ) ).

% mono_SucI2
tff(fact_8582_mono__SucI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N2))),aa_nat_rat(X6,N2)))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI2
tff(fact_8583_mono__SucI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N2))),aa_nat_num(X6,N2)))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI2
tff(fact_8584_mono__SucI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N2))),aa_nat_nat(X6,N2)))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI2
tff(fact_8585_mono__SucI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N2))),aa_nat_int(X6,N2)))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI2
tff(fact_8586_mono__SucI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,N2)),aa_nat_real(X6,aa_nat_nat(suc,N2))))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI1
tff(fact_8587_mono__SucI1,axiom,
    ! [X6: fun_nat_set_int] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(X6,N2)),aa_nat_set_int(X6,aa_nat_nat(suc,N2))))
     => topolo3100542954746470799et_int(X6) ) ).

% mono_SucI1
tff(fact_8588_mono__SucI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N2)),aa_nat_rat(X6,aa_nat_nat(suc,N2))))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI1
tff(fact_8589_mono__SucI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X6,N2)),aa_nat_num(X6,aa_nat_nat(suc,N2))))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI1
tff(fact_8590_mono__SucI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X6,N2)),aa_nat_nat(X6,aa_nat_nat(suc,N2))))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI1
tff(fact_8591_mono__SucI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X6,N2)),aa_nat_int(X6,aa_nat_nat(suc,N2))))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI1
tff(fact_8592_gbinomial__code,axiom,
    ! [K: nat,A: complex] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = one_one_complex ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_zw(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K)) ) ) ) ).

% gbinomial_code
tff(fact_8593_gbinomial__code,axiom,
    ! [K: nat,A: rat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = one_one_rat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_zx(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ) ) ).

% gbinomial_code
tff(fact_8594_gbinomial__code,axiom,
    ! [K: nat,A: real] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = one_one_real ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_zy(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K)) ) ) ) ).

% gbinomial_code
tff(fact_8595_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( ( 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))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
          & ( ~ 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))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_8596_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = 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,K)),aa_nat_int(semiri1314217659103216013at_int,M))),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,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = 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,K)),aa_nat_int(semiri1314217659103216013at_int,M))),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,M),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != 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,K)),aa_nat_int(semiri1314217659103216013at_int,M))),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),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_8597_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = 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,K)),aa_nat_int(semiri1314217659103216013at_int,M))),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,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = 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,K)),aa_nat_int(semiri1314217659103216013at_int,M))),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,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != 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,K)),aa_nat_int(semiri1314217659103216013at_int,M))),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,M),N)),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_8598_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,L),numera6620942414471956472nteger(aa_num_num(bit1,K))) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(aa_num_nat(pred_numeral,L),numera6620942414471956472nteger(K)),numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).

% take_bit_numeral_bit1
tff(fact_8599_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_8600_take__bit__numeral__bit1,axiom,
    ! [L: num,K: 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,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_8601_sgn__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(sgn_sgn_int,A)) = aa_int_int(sgn_sgn_int,A) ).

% sgn_sgn
tff(fact_8602_sgn__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% sgn_sgn
tff(fact_8603_sgn__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_rat_rat(sgn_sgn_rat,A) ).

% sgn_sgn
tff(fact_8604_sgn__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(sgn_sgn_complex,A)) = aa_complex_complex(sgn_sgn_complex,A) ).

% sgn_sgn
tff(fact_8605_sgn__sgn,axiom,
    ! [A: code_integer] : sgn_sgn_Code_integer(sgn_sgn_Code_integer(A)) = sgn_sgn_Code_integer(A) ).

% sgn_sgn
tff(fact_8606_sgn__0,axiom,
    sgn_sgn_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_8607_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_8608_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_8609_sgn__0,axiom,
    aa_rat_rat(sgn_sgn_rat,zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_8610_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_8611_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_8612_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_8613_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_8614_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_8615_sgn__1,axiom,
    aa_rat_rat(sgn_sgn_rat,one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_8616_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_8617_sgn__1,axiom,
    sgn_sgn_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_8618_sgn__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B)) ).

% sgn_divide
tff(fact_8619_sgn__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_divide
tff(fact_8620_sgn__divide,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_divide
tff(fact_8621_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_8622_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_8623_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_8624_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_8625_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] : sgn_sgn_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,sgn_sgn_Code_integer(A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_8626_sgn__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,sgn_sgn_Code_integer(A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% sgn_less
tff(fact_8627_sgn__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_less
tff(fact_8628_sgn__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% sgn_less
tff(fact_8629_sgn__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_less
tff(fact_8630_sgn__greater,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),sgn_sgn_Code_integer(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% sgn_greater
tff(fact_8631_sgn__greater,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_greater
tff(fact_8632_sgn__greater,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% sgn_greater
tff(fact_8633_sgn__greater,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_greater
tff(fact_8634_divide__sgn,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(sgn_sgn_rat,B)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(sgn_sgn_rat,B)) ).

% divide_sgn
tff(fact_8635_divide__sgn,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(sgn_sgn_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(sgn_sgn_real,B)) ).

% divide_sgn
tff(fact_8636_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_8637_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_8638_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_8639_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_8640_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_8641_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_8642_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_8643_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_8644_sgn__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( sgn_sgn_Code_integer(A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_8645_sgn__pos,axiom,
    ! [A: 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 ) ) ).

% sgn_pos
tff(fact_8646_sgn__pos,axiom,
    ! [A: 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 ) ) ).

% sgn_pos
tff(fact_8647_sgn__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_8648_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_8649_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_8650_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_8651_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_8652_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_8653_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_8654_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_8655_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_8656_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_8657_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(sgn_sgn_Code_integer(A),sgn_sgn_Code_integer(A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% sgn_mult_self_eq
tff(fact_8658_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_mult_self_eq
tff(fact_8659_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% sgn_mult_self_eq
tff(fact_8660_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_mult_self_eq
tff(fact_8661_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : sgn_sgn_Code_integer(abs_abs_Code_integer(A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8662_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,A),zero_zero_complex))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8663_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8664_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(abs_abs_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8665_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_8666_sgn__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,A),zero_z3403309356797280102nteger))) ).

% sgn_abs
tff(fact_8667_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = zero_n1201886186963655149omplex(aa_bool_bool(fNot,aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,A),zero_zero_complex))) ).

% sgn_abs
tff(fact_8668_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(aa_bool_bool(fNot,aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,A),zero_zero_real))) ).

% sgn_abs
tff(fact_8669_sgn__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,A),zero_zero_rat))) ).

% sgn_abs
tff(fact_8670_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,A),zero_zero_int))) ).

% sgn_abs
tff(fact_8671_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_8672_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% less_numeral_Suc
tff(fact_8673_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_8674_le__Suc__numeral,axiom,
    ! [N: nat,K: 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,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_8675_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% le_numeral_Suc
tff(fact_8676_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_8677_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_8678_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: 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,K),aa_int_int(sgn_sgn_int,R2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_8679_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: 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)),K)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_8680_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: 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))),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_8681_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: 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)),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_8682_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),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,K)),N)) ).

% max_numeral_Suc
tff(fact_8683_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_8684_sgn__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_8685_sgn__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_8686_sgn__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_8687_sgn__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( sgn_sgn_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_8688_sgn__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = zero_n2052037380579107095ol_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_8689_sgn__of__nat,axiom,
    ! [N: nat] : sgn_sgn_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_8690_sgn__of__nat,axiom,
    ! [N: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,N)) = zero_n3304061248610475627l_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_8691_sgn__of__nat,axiom,
    ! [N: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% sgn_of_nat
tff(fact_8692_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,N) != zero_zero_complex ).

% fact_nonzero
tff(fact_8693_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) != zero_zero_rat ).

% fact_nonzero
tff(fact_8694_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).

% fact_nonzero
tff(fact_8695_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).

% fact_nonzero
tff(fact_8696_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) != zero_zero_real ).

% fact_nonzero
tff(fact_8697_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_8698_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_8699_sgn__zero__iff,axiom,
    ! [X2: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,X2) = zero_zero_complex )
    <=> ( X2 = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_8700_sgn__zero__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(sgn_sgn_real,X2) = zero_zero_real )
    <=> ( X2 = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_8701_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_8702_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_8703_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_8704_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_8705_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_8706_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_8707_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_8708_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_8709_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_8710_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( sgn_sgn_Code_integer(B) = sgn_sgn_Code_integer(A) )
     => ( sgn_sgn_Code_integer(plus_p5714425477246183910nteger(A,B)) = sgn_sgn_Code_integer(A) ) ) ).

% same_sgn_sgn_add
tff(fact_8711_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_8712_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(sgn_sgn_rat,A) ) ) ).

% same_sgn_sgn_add
tff(fact_8713_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_8714_sgn__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_mult
tff(fact_8715_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : sgn_sgn_Code_integer(times_3573771949741848930nteger(A,B)) = times_3573771949741848930nteger(sgn_sgn_Code_integer(A),sgn_sgn_Code_integer(B)) ).

% sgn_mult
tff(fact_8716_sgn__mult,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_mult
tff(fact_8717_sgn__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B)) ).

% sgn_mult
tff(fact_8718_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_8719_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_8720_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_8721_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_8722_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) != aa_rat_rat(sgn_sgn_rat,A) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_8723_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( sgn_sgn_Code_integer(B) != sgn_sgn_Code_integer(A) )
     => ( ( sgn_sgn_Code_integer(A) != zero_z3403309356797280102nteger )
       => ( ( sgn_sgn_Code_integer(B) != zero_z3403309356797280102nteger )
         => ( sgn_sgn_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,sgn_sgn_Code_integer(B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_8724_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_8725_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_8726_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_8727_sgn__minus__1,axiom,
    aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% sgn_minus_1
tff(fact_8728_sgn__minus__1,axiom,
    sgn_sgn_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_8729_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_zero
tff(fact_8730_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_zero
tff(fact_8731_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_zero
tff(fact_8732_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_zero
tff(fact_8733_mult__sgn__abs,axiom,
    ! [X2: code_integer] : times_3573771949741848930nteger(sgn_sgn_Code_integer(X2),abs_abs_Code_integer(X2)) = X2 ).

% mult_sgn_abs
tff(fact_8734_mult__sgn__abs,axiom,
    ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X2)),aa_real_real(abs_abs_real,X2)) = X2 ).

% mult_sgn_abs
tff(fact_8735_mult__sgn__abs,axiom,
    ! [X2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,X2)),aa_rat_rat(abs_abs_rat,X2)) = X2 ).

% mult_sgn_abs
tff(fact_8736_mult__sgn__abs,axiom,
    ! [X2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X2)),aa_int_int(abs_abs_int,X2)) = X2 ).

% mult_sgn_abs
tff(fact_8737_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_8738_sgn__mult__abs,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(sgn_sgn_Code_integer(A),abs_abs_Code_integer(A)) = A ).

% sgn_mult_abs
tff(fact_8739_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_8740_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(abs_abs_rat,A)) = A ).

% sgn_mult_abs
tff(fact_8741_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_8742_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_8743_abs__mult__sgn,axiom,
    ! [A: code_integer] : times_3573771949741848930nteger(abs_abs_Code_integer(A),sgn_sgn_Code_integer(A)) = A ).

% abs_mult_sgn
tff(fact_8744_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_8745_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = A ).

% abs_mult_sgn
tff(fact_8746_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_8747_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : abs_abs_Code_integer(K) = times_3573771949741848930nteger(K,sgn_sgn_Code_integer(K)) ).

% linordered_idom_class.abs_sgn
tff(fact_8748_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: real] : aa_real_real(abs_abs_real,K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),aa_real_real(sgn_sgn_real,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_8749_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : aa_rat_rat(abs_abs_rat,K) = aa_rat_rat(times_times_rat(K),aa_rat_rat(sgn_sgn_rat,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_8750_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_8751_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_8752_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_8753_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_8754_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_8755_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_8756_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_8757_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_8758_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),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,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_8759_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_8760_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_8761_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),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,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_8762_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = 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_cg),set_or1269000886237332187st_nat(aa_nat_nat(suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8763_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = 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,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8764_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cg),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_8765_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))
     => ? [B8: 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_zz(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_8766_sin__coeff__def,axiom,
    ! [X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X))
       => ( aa_nat_real(sin_coeff,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))),X))
       => ( aa_nat_real(sin_coeff,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X)) ) ) ) ).

% sin_coeff_def
tff(fact_8767_Maclaurin__exp__lt,axiom,
    ! [X2: real,N: nat] :
      ( ( X2 != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X2)))
            & ( aa_real_real(exp_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aaa(X2)),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,T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8768_sin__paired,axiom,
    ! [X2: real] : pp(aa_real_bool(sums_real(aTP_Lamm_aab(X2)),aa_real_real(sin_real,X2))) ).

% sin_paired
tff(fact_8769_exp__less__mono,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X2)),aa_real_real(exp_real,Y))) ) ).

% exp_less_mono
tff(fact_8770_exp__less__cancel__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X2)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% exp_less_cancel_iff
tff(fact_8771_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8772_one__less__exp__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% one_less_exp_iff
tff(fact_8773_exp__less__one__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X2)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_8774_exp__ln,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X2)) = X2 ) ) ).

% exp_ln
tff(fact_8775_exp__ln__iff,axiom,
    ! [X2: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X2)) = X2 )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% exp_ln_iff
tff(fact_8776_exp__less__cancel,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X2)),aa_real_real(exp_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% exp_less_cancel
tff(fact_8777_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_8778_exp__gt__zero,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X2))) ).

% exp_gt_zero
tff(fact_8779_not__exp__less__zero,axiom,
    ! [X2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X2)),zero_zero_real)) ).

% not_exp_less_zero
tff(fact_8780_exp__gt__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X2))) ) ).

% exp_gt_one
tff(fact_8781_sin__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X2))) ) ) ).

% sin_gt_zero
tff(fact_8782_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_8783_ln__ge__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X2)) ) ) ).

% ln_ge_iff
tff(fact_8784_sin__eq__0__pi,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),pi))
       => ( ( aa_real_real(sin_real,X2) = zero_zero_real )
         => ( X2 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8785_sin__zero__pi__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),pi))
     => ( ( aa_real_real(sin_real,X2) = zero_zero_real )
      <=> ( X2 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8786_sin__gt__zero__02,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2))) ) ) ).

% sin_gt_zero_02
tff(fact_8787_sgn__power__injE,axiom,
    ! [A: real,N: nat,X2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)) = X2 )
     => ( ( X2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_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_8788_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_8789_sin__gt__zero2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X2))) ) ) ).

% sin_gt_zero2
tff(fact_8790_sin__lt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_8791_sin__le__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_8792_sin__less__zero,axiom,
    ! [X2: 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)))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X2)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_8793_sin__mono__less__eq,axiom,
    ! [X2: 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))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( 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,X2)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8794_sin__monotone__2pi,axiom,
    ! [Y: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X2))) ) ) ) ).

% sin_monotone_2pi
tff(fact_8795_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X2))
     => ( 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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X2))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_8796_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_8797_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X2),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X2)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_8798_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),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_8799_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: 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)))
       => ( ( K = 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) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q2,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8800_Maclaurin__sin__expansion4,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),X2))
          & ( aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aac(X2)),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8801_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X2: 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),X2))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X2))
            & ( aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aac(X2)),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8802_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q4: int] :
              ( ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = product_Pair_int_int(Q4,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,Q4),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8803_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K2) ) )
        | ? [L4: int,K2: int,Q5: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4) ) )
        | ? [R5: int,L4: int,K2: int,Q5: int] :
            ( ( A1 = K2 )
            & ( 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)))
            & ( K2 = 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_8804_sincos__total__2pi,axiom,
    ! [X2: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X2 = aa_real_real(cos_real,T6) )
               => ( Y != aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8805_sin__tan,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(sin_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X2)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8806_cos__monotone__0__pi,axiom,
    ! [Y: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X2)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8807_cos__mono__less__eq,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8808_forall__pos__mono__1,axiom,
    ! [P: 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(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_8809_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N3: nat] :
          ( ( N3 != 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,N3))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))),E2)) ) ) ).

% real_arch_inverse
tff(fact_8810_forall__pos__mono,axiom,
    ! [P: 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(P,D4))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_8811_ln__inverse,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X2)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X2)) ) ) ).

% ln_inverse
tff(fact_8812_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_8813_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X2))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8814_sincos__principal__value,axiom,
    ! [X2: real] :
    ? [Y2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),pi))
      & ( aa_real_real(sin_real,Y2) = aa_real_real(sin_real,X2) )
      & ( aa_real_real(cos_real,Y2) = aa_real_real(cos_real,X2) ) ) ).

% sincos_principal_value
tff(fact_8815_cos__tan,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(cos_real,X2) = 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(power_power_real(aa_real_real(tan_real,X2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8816_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_8817_tan__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X2))) ) ) ).

% tan_gt_zero
tff(fact_8818_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 )
      & ! [Y3: 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))))),Y3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),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,Y3) = Y ) )
         => ( Y3 = X4 ) ) ) ).

% tan_total
tff(fact_8819_tan__monotone,axiom,
    ! [Y: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X2))) ) ) ) ).

% tan_monotone
tff(fact_8820_tan__monotone_H,axiom,
    ! [Y: real,X2: 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))))),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X2))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8821_tan__mono__lt__eq,axiom,
    ! [X2: 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))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( 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,X2)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8822_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_8823_cos__double__less__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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))),X2))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_8824_cos__gt__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X2))) ) ) ).

% cos_gt_zero
tff(fact_8825_plus__inverse__ge__2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => 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,X2),aa_real_real(inverse_inverse_real,X2)))) ) ).

% plus_inverse_ge_2
tff(fact_8826_real__inv__sqrt__pow2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X2) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8827_tan__pos__pi2__le,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X2))) ) ) ).

% tan_pos_pi2_le
tff(fact_8828_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_8829_tan__less__zero,axiom,
    ! [X2: 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)))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X2)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_8830_tan__mono__le__eq,axiom,
    ! [X2: 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))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( 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,X2)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8831_tan__mono__le,axiom,
    ! [X2: 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))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_8832_tan__bound__pi2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),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,X2))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_8833_cos__gt__zero__pi,axiom,
    ! [X2: 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))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X2))) ) ) ).

% cos_gt_zero_pi
tff(fact_8834_arctan__unique,axiom,
    ! [X2: 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))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( ( aa_real_real(tan_real,X2) = Y )
         => ( aa_real_real(arctan,Y) = X2 ) ) ) ) ).

% arctan_unique
tff(fact_8835_arctan__tan,axiom,
    ! [X2: 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))))),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X2)) = X2 ) ) ) ).

% arctan_tan
tff(fact_8836_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_8837_tan__total__pi4,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),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) = X2 ) ) ) ).

% tan_total_pi4
tff(fact_8838_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X2: 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,X2),zero_zero_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
            & ( aa_real_real(cos_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aad(X2)),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8839_Maclaurin__cos__expansion2,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X2))
            & ( aa_real_real(cos_real,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aad(X2)),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,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8840_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T6)),aa_real_real(sin_real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8841_sinh__real__less__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X2)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ).

% sinh_real_less_iff
tff(fact_8842_sinh__real__neg__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X2)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ).

% sinh_real_neg_iff
tff(fact_8843_sinh__real__pos__iff,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2)) ) ).

% sinh_real_pos_iff
tff(fact_8844_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8845_sinh__less__cosh__real,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X2)),aa_real_real(cosh_real,X2))) ).

% sinh_less_cosh_real
tff(fact_8846_cosh__real__pos,axiom,
    ! [X2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X2))) ).

% cosh_real_pos
tff(fact_8847_cosh__real__strict__mono,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X2)),aa_real_real(cosh_real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_8848_cosh__real__nonneg__less__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( 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,X2)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_8849_cosh__real__nonpos__less__iff,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_8850_cosh__ln__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),aa_real_real(inverse_inverse_real,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8851_sinh__ln__real,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(inverse_inverse_real,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8852_log__base__10__eq1,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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))))),X2) = 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,X2)) ) ) ).

% log_base_10_eq1
tff(fact_8853_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_8854_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aae,Y)) ).

% arctan_def
tff(fact_8855_log__base__10__eq2,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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))))),X2) = 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,X2)) ) ) ).

% log_base_10_eq2
tff(fact_8856_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_8857_nat__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).

% nat_int
tff(fact_8858_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_8859_nat__of__bool,axiom,
    ! [P: bool] : aa_int_nat(nat2,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n2687167440665602831ol_nat(P) ).

% nat_of_bool
tff(fact_8860_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_8861_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_8862_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_8863_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_8864_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_8865_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_8866_log__less__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( 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),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_8867_log__less__one__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_8868_one__less__log__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_8869_log__less__zero__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_8870_zero__less__log__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_8871_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_8872_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_8873_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_8874_log__le__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( 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),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_8875_log__le__one__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),A)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_8876_one__le__log__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_8877_log__le__zero__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_8878_zero__le__log__cancel__iff,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_8879_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_8880_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X2)),N) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_8881_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X2: 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,X2)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X2)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_8882_nat__abs__dvd__iff,axiom,
    ! [K: 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,K))),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_abs_dvd_iff
tff(fact_8883_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_8884_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_8885_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(power_power_real(A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_8886_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_8887_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2)),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,X2)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_8888_numeral__power__less__nat__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2)),N)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_8889_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: 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,X2)),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,X2)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_8890_numeral__power__le__nat__cancel__iff,axiom,
    ! [X2: 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,X2)),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,X2)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_8891_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_8892_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_8893_nat__numeral__as__int,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,X) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X)) ).

% nat_numeral_as_int
tff(fact_8894_nat__mono,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X2)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_8895_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: 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),Z5))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_8896_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% all_nat
tff(fact_8897_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% ex_nat
tff(fact_8898_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_8899_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_8900_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_8901_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_8902_nat__le__iff,axiom,
    ! [X2: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X2)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_8903_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_8904_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = 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_8905_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_8906_int__minus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_8907_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_8908_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_8909_nat__times__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_8910_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_8911_nat__minus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_8912_nat__div__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_8913_nat__mod__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_8914_log__base__change,axiom,
    ! [A: real,B: real,X2: 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),X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_8915_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_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,M)) ) ) ) ).

% log_of_power_eq
tff(fact_8916_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),N)),M))
     => ( 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),M))) ) ) ).

% less_log_of_power
tff(fact_8917_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_8918_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_8919_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_8920_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = 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,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_8921_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N3: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => pp(aa_nat_bool(P,N3)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_8922_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).

% le_nat_iff
tff(fact_8923_nat__add__distrib,axiom,
    ! [Z: int,Z5: 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),Z5))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_8924_nat__mult__distrib,axiom,
    ! [Z: int,Z5: 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),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_8925_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_8926_nat__diff__distrib_H,axiom,
    ! [X2: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( 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,X2),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X2)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_8927_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z5),Z))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_8928_nat__abs__triangle__ineq,axiom,
    ! [K: 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,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_8929_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_8930_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_8931_log__mult,axiom,
    ! [A: real,X2: 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),X2))
         => ( 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,X2),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_8932_log__divide,axiom,
    ! [A: real,X2: 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),X2))
         => ( 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,X2),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_8933_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),N)),M))
     => ( 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),M))) ) ) ).

% le_log_of_power
tff(fact_8934_log__base__pow,axiom,
    ! [A: real,N: nat,X2: 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(power_power_real(A),N)),X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_8935_log__nat__power,axiom,
    ! [X2: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(power_power_real(X2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)) ) ) ).

% log_nat_power
tff(fact_8936_log__inverse,axiom,
    ! [A: real,X2: 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),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,X2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)) ) ) ) ) ).

% log_inverse
tff(fact_8937_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_8938_nat__less__iff,axiom,
    ! [W2: int,M: 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)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_8939_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: 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),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_8940_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_8941_diff__nat__eq__if,axiom,
    ! [Z5: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = 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),Z5)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5))) ) ) ) ).

% diff_nat_eq_if
tff(fact_8942_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_8943_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_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),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_8944_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X2: 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),X2))
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),X2) = 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),X2)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8945_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M))
    <=> ( ( 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,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_8946_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_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),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_8947_less__log2__of__power,axiom,
    ! [N: nat,M: 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)),M))
     => 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,M)))) ) ).

% less_log2_of_power
tff(fact_8948_le__log2__of__power,axiom,
    ! [N: nat,M: 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)),M))
     => 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,M)))) ) ).

% le_log2_of_power
tff(fact_8949_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( 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_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_8950_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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),M))
       => 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,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_8951_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: 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),K))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,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)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_8952_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: 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)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( 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,K))) = 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_8953_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_8954_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: 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),K))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,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)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8955_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: 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)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( 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,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8956_nat__ceiling__le__eq,axiom,
    ! [X2: 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,X2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_8957_nat__floor__neg,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X2)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_8958_floor__eq3,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X2)) = N ) ) ) ).

% floor_eq3
tff(fact_8959_le__nat__floor,axiom,
    ! [X2: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X2)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_8960_floor__eq,axiom,
    ! [N: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2) = N ) ) ) ).

% floor_eq
tff(fact_8961_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_8962_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_8963_floor__eq4,axiom,
    ! [N: nat,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X2)) = N ) ) ) ).

% floor_eq4
tff(fact_8964_floor__eq2,axiom,
    ! [N: int,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2) = N ) ) ) ).

% floor_eq2
tff(fact_8965_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_8966_ceiling__log__eq__powr__iff,axiom,
    ! [X2: real,B: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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),X2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),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,K))),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8967_powr__int,axiom,
    ! [X2: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(power_power_real(X2),aa_int_nat(nat2,I)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(X2),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_8968_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_8969_floor__log__eq__powr__iff,axiom,
    ! [X2: real,B: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( 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),X2)) = K )
        <=> ( 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,K))),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8970_powr__gt__zero,axiom,
    ! [X2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A)))
    <=> ( X2 != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8971_powr__less__cancel__iff,axiom,
    ! [X2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel_iff
tff(fact_8972_powr__eq__one__iff,axiom,
    ! [A: real,X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X2) = one_one_real )
      <=> ( X2 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8973_powr__le__cancel__iff,axiom,
    ! [X2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% powr_le_cancel_iff
tff(fact_8974_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_8975_powr__log__cancel,axiom,
    ! [A: real,X2: 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),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),X2)) = X2 ) ) ) ) ).

% powr_log_cancel
tff(fact_8976_powr__less__mono2__neg,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2),A))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_8977_powr__non__neg,axiom,
    ! [A: real,X2: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X2)),zero_zero_real)) ).

% powr_non_neg
tff(fact_8978_powr__less__mono,axiom,
    ! [A: real,B: real,X2: 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),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B))) ) ) ).

% powr_less_mono
tff(fact_8979_powr__less__cancel,axiom,
    ! [X2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel
tff(fact_8980_powr__less__mono2,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).

% powr_less_mono2
tff(fact_8981_powr__mono2_H,axiom,
    ! [A: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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,X2),A))) ) ) ) ).

% powr_mono2'
tff(fact_8982_powr__inj,axiom,
    ! [A: real,X2: 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),X2) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( X2 = Y ) ) ) ) ).

% powr_inj
tff(fact_8983_gr__one__powr,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( 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,X2),Y))) ) ) ).

% gr_one_powr
tff(fact_8984_powr__realpow,axiom,
    ! [X2: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(power_power_real(X2),N) ) ) ).

% powr_realpow
tff(fact_8985_powr__less__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X2))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2))) ) ) ) ).

% powr_less_iff
tff(fact_8986_less__powr__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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),X2)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_8987_log__less__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_less_iff
tff(fact_8988_less__log__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X2)) ) ) ) ).

% less_log_iff
tff(fact_8989_powr__neg__one,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X2) ) ) ).

% powr_neg_one
tff(fact_8990_arcsin__less__arcsin,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8991_arcsin__less__mono,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),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,X2)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_8992_powr__le__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X2))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2))) ) ) ) ).

% powr_le_iff
tff(fact_8993_le__powr__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),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),X2)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_8994_log__le__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_le_iff
tff(fact_8995_le__log__iff,axiom,
    ! [B: real,X2: 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),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X2)) ) ) ) ).

% le_log_iff
tff(fact_8996_ln__powr__bound,axiom,
    ! [X2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X2))
     => ( 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,X2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),A)),A))) ) ) ).

% ln_powr_bound
tff(fact_8997_ln__powr__bound2,axiom,
    ! [X2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => ( 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,X2)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),X2))) ) ) ).

% ln_powr_bound2
tff(fact_8998_log__add__eq__powr,axiom,
    ! [B: real,X2: 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),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8999_add__log__eq__powr,axiom,
    ! [B: real,X2: 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),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)) = 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)),X2)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_9000_minus__log__eq__powr,axiom,
    ! [B: real,X2: 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),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)) = 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)),X2)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_9001_cos__arcsin__nonzero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X2)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9002_log__minus__eq__powr,axiom,
    ! [B: real,X2: 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),X2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9003_powr__neg__numeral,axiom,
    ! [X2: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_9004_powr__real__of__int,axiom,
    ! [X2: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => ( ( 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,X2),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(power_power_real(X2),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,X2),aa_int_real(ring_1_of_int_real,N)) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(X2),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_9005_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),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)))),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_9006_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),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)))),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_9007_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_9008_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_9009_xor__nat__numerals_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2)) ).

% xor_nat_numerals(3)
tff(fact_9010_xor__nat__numerals_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2)) ).

% xor_nat_numerals(4)
tff(fact_9011_floor__real__def,axiom,
    ! [X2: real] : aa_real_int(archim6058952711729229775r_real,X2) = the_int(aa_real_fun_int_bool(aTP_Lamm_aaf,X2)) ).

% floor_real_def
tff(fact_9012_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),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,M),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,M),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_9013_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),M))),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,M),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_9014_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_9015_floor__rat__def,axiom,
    ! [X2: rat] : aa_rat_int(archim3151403230148437115or_rat,X2) = the_int(aTP_Lamm_aag(X2)) ).

% floor_rat_def
tff(fact_9016_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),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,M),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,M),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_9017_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_9018_or__nat__numerals_I4_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2)) ).

% or_nat_numerals(4)
tff(fact_9019_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_9020_or__nat__numerals_I3_J,axiom,
    ! [X2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X2))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X2)) ).

% or_nat_numerals(3)
tff(fact_9021_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_9022_less__eq__rat__def,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
        | ( X2 = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9023_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))
           => ! [T6: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T6))
               => ( R2 != aa_rat_rat(plus_plus_rat(S3),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9024_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_9025_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_9026_XOR__upper,axiom,
    ! [X2: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),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(X2),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_9027_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),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_9028_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),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_9029_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),M))),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,M),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_9030_normalize__negative,axiom,
    ! [Q2: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q2),zero_zero_int))
     => ( normalize(product_Pair_int_int(P3,Q2)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9031_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),set_or4665077453230672383an_nat(M,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,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9032_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,aa_VEB5832953974749077948T_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_9033_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M4))
         => ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N3))
             => 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,M4)),aa_nat_real(X6,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_9034_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),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_9035_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or4665077453230672383an_nat(L,U))) ).

% finite_atLeastLessThan
tff(fact_9036_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9037_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
          & pp(aa_nat_bool(P,M4)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_9038_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
         => pp(aa_nat_bool(P,M4)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_9039_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9040_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_or4665077453230672383an_nat(zero_zero_nat,N) ).

% lessThan_atLeast0
tff(fact_9041_atLeastLessThan0,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9042_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% atLeast0_lessThan_Suc
tff(fact_9043_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),set_or4665077453230672383an_nat(zero_zero_nat,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9044_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = insert_nat(N,set_or4665077453230672383an_nat(M,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_9045_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_9046_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_9047_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q2: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).

% normalize_denom_pos
tff(fact_9048_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(M,aa_num_nat(pred_numeral,K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_9049_OR__upper,axiom,
    ! [X2: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),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(X2),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_9050_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: 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_9051_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9052_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_aah(A),B)),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),set_or4665077453230672383an_nat(zero_zero_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9053_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,aa_VEB5832953974749077948T_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_9054_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(L,U))) ).

% finite_atLeastLessThan_int
tff(fact_9055_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_9056_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9057_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_9058_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_9059_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,N))) = N ).

% card_Collect_less_nat
tff(fact_9060_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_9061_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or4665077453230672383an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_9062_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_az,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_9063_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or1269000886237332187st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9064_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_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_9065_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_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_9066_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_9067_card__less__Suc2,axiom,
    ! [M5: set_nat,I: 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_aai(M5),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aaj(M5),I))) ) ) ).

% card_less_Suc2
tff(fact_9068_card__less__Suc,axiom,
    ! [M5: set_nat,I: 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_aai(M5),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aaj(M5),I))) ) ) ).

% card_less_Suc
tff(fact_9069_card__less,axiom,
    ! [M5: set_nat,I: 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_aaj(M5),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9070_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2)))))
     => ( A2 = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9071_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),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_9072_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_cg),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),S2))) ).

% card_sum_le_nat_sum
tff(fact_9073_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != 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_aak(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_9074_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_cw,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_9075_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: 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),K)),zero_zero_int))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_9076_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_9077_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_9078_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_9079_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),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,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9080_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M2))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_9081_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_bm),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_9082_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_aal),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_9083_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_9084_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_aam,N)),M) ).

% max_Suc1
tff(fact_9085_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_aan,N)),M) ).

% max_Suc2
tff(fact_9086_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cg),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_9087_binomial__def,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_aao(N),K))) ).

% binomial_def
tff(fact_9088_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: 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),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_9089_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_9090_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: 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,M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_9091_bit__push__bit__iff__nat,axiom,
    ! [M: 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,M),Q2)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),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),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9092_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cg),Nat) ).

% pred_def
tff(fact_9093_bezw__0,axiom,
    ! [X2: nat] : bezw(X2,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_9094_prod__decode__aux_Oelims,axiom,
    ! [X2: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X2,Xa2) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X2))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),Xa2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X2))
         => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X2))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_9095_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(K,M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_9096_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_9097_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9098_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_9099_root__powr__inverse,axiom,
    ! [N: nat,X2: 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),X2))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_9100_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: 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),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_9101_real__root__Suc__0,axiom,
    ! [X2: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X2) = X2 ).

% real_root_Suc_0
tff(fact_9102_root__0,axiom,
    ! [X2: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X2) = zero_zero_real ).

% root_0
tff(fact_9103_real__root__eq__iff,axiom,
    ! [N: nat,X2: 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),X2) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X2 = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9104_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)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_9105_real__root__eq__0__iff,axiom,
    ! [N: nat,X2: 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),X2) = zero_zero_real )
      <=> ( X2 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9106_real__root__less__iff,axiom,
    ! [N: nat,X2: 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),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y)) ) ) ).

% real_root_less_iff
tff(fact_9107_real__root__le__iff,axiom,
    ! [N: nat,X2: 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),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y)) ) ) ).

% real_root_le_iff
tff(fact_9108_real__root__eq__1__iff,axiom,
    ! [N: nat,X2: 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),X2) = one_one_real )
      <=> ( X2 = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9109_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_9110_real__root__lt__0__iff,axiom,
    ! [N: nat,X2: 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),X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_9111_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_9112_real__root__le__0__iff,axiom,
    ! [N: nat,X2: 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),X2)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_9113_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_9114_real__root__lt__1__iff,axiom,
    ! [N: nat,X2: 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),X2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_9115_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_9116_real__root__le__1__iff,axiom,
    ! [N: nat,X2: 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),X2)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_9117_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_9118_real__root__pow__pos2,axiom,
    ! [N: nat,X2: 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),X2))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),N) = X2 ) ) ) ).

% real_root_pow_pos2
tff(fact_9119_real__root__less__mono,axiom,
    ! [N: nat,X2: 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,X2),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_9120_real__root__le__mono,axiom,
    ! [N: nat,X2: 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,X2),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_9121_real__root__power,axiom,
    ! [N: nat,X2: real,K: 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(power_power_real(X2),K)) = aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),K) ) ) ).

% real_root_power
tff(fact_9122_real__root__abs,axiom,
    ! [N: nat,X2: 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,X2)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)) ) ) ).

% real_root_abs
tff(fact_9123_sgn__root,axiom,
    ! [N: nat,X2: 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),X2)) = aa_real_real(sgn_sgn_real,X2) ) ) ).

% sgn_root
tff(fact_9124_real__root__gt__zero,axiom,
    ! [N: nat,X2: 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),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ).

% real_root_gt_zero
tff(fact_9125_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X2: 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),X2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),X2)),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9126_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(power_power_real(Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9127_real__root__pos__pos,axiom,
    ! [N: nat,X2: 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),X2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ).

% real_root_pos_pos
tff(fact_9128_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X2: 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),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N4),X2))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9129_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X2: 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),X2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),X2)),aa_real_real(aa_nat_fun_real_real(root,N),X2))) ) ) ) ).

% real_root_decreasing
tff(fact_9130_real__root__pow__pos,axiom,
    ! [N: nat,X2: 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),X2))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),N) = X2 ) ) ) ).

% real_root_pow_pos
tff(fact_9131_real__root__power__cancel,axiom,
    ! [N: nat,X2: 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),X2))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(power_power_real(X2),N)) = X2 ) ) ) ).

% real_root_power_cancel
tff(fact_9132_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X2: 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(power_power_real(Y),N) = X2 )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9133_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X2: 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),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_real_real(aa_nat_fun_real_real(root,N4),X2))) ) ) ) ) ).

% real_root_increasing
tff(fact_9134_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(power_power_real(aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_9135_sgn__power__root,axiom,
    ! [N: nat,X2: 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),X2))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X2))),N)) = X2 ) ) ).

% sgn_power_root
tff(fact_9136_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_9137_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_9138_log__base__root,axiom,
    ! [N: nat,B: real,X2: 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)),X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X2)) ) ) ) ).

% log_base_root
tff(fact_9139_split__root,axiom,
    ! [P: fun_real_bool,N: nat,X2: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X2)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y4: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y4)),N)) = X2 )
             => pp(aa_real_bool(P,Y4)) ) ) ) ) ).

% split_root
tff(fact_9140_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_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_9141_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or5832277885323065728an_int(L,U))) ).

% finite_greaterThanLessThan_int
tff(fact_9142_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9143_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_az,aTP_Lamm_ay) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9144_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or5834768355832116004an_nat(L,U))) ).

% finite_greaterThanLessThan
tff(fact_9145_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_9146_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or5834768355832116004an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9147_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9148_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: 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,X2)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_aaq),Xa2),X2)) ).

% times_int.abs_eq
tff(fact_9149_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_9150_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y2: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y2)) ).

% eq_Abs_Integ
tff(fact_9151_Gcd__nat__eq__one,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(one_one_nat),N4))
     => ( aa_set_nat_nat(gcd_Gcd_nat,N4) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9152_Gcd__dvd__nat,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(gcd_Gcd_nat,A2)),A)) ) ).

% Gcd_dvd_nat
tff(fact_9153_Gcd__greatest__nat,axiom,
    ! [A2: set_nat,A: nat] :
      ( ! [B3: nat] :
          ( pp(aa_set_nat_bool(member_nat(B3),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B3)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_set_nat_nat(gcd_Gcd_nat,A2))) ) ).

% Gcd_greatest_nat
tff(fact_9154_int_Oabs__induct,axiom,
    ! [P: fun_int_bool,X2: int] :
      ( ! [Y2: product_prod_nat_nat] : pp(aa_int_bool(P,aa_Pro4240683299535347697at_int(abs_Integ,Y2)))
     => pp(aa_int_bool(P,X2)) ) ).

% int.abs_induct
tff(fact_9155_nat_Oabs__eq,axiom,
    ! [X2: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X2)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X2) ).

% nat.abs_eq
tff(fact_9156_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_9157_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_9158_uminus__int_Oabs__eq,axiom,
    ! [X2: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X2)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_aar),X2)) ).

% uminus_int.abs_eq
tff(fact_9159_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_9160_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: 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,X2)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_aat),Xa2),X2)) ) ).

% less_int.abs_eq
tff(fact_9161_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: 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,X2)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_aav),Xa2),X2)) ) ).

% less_eq_int.abs_eq
tff(fact_9162_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: 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,X2)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_aax),Xa2),X2)) ).

% plus_int.abs_eq
tff(fact_9163_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: 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,X2)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_aaz),Xa2),X2)) ).

% minus_int.abs_eq
tff(fact_9164_abs__Gcd__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Gcd_int(K5)) = gcd_Gcd_int(K5) ).

% abs_Gcd_eq
tff(fact_9165_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_9166_Gcd__dvd__int,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,gcd_Gcd_int(A2)),A)) ) ).

% Gcd_dvd_int
tff(fact_9167_Gcd__greatest__int,axiom,
    ! [A2: set_int,A: int] :
      ( ! [B3: int] :
          ( pp(aa_set_int_bool(member_int(B3),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B3)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),gcd_Gcd_int(A2))) ) ).

% Gcd_greatest_int
tff(fact_9168_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5))) ).

% Gcd_int_greater_eq_0
tff(fact_9169_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).

% eval_nat_numeral(2)
tff(fact_9170_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_9171_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,N)),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_9172_sub__BitM__One__eq,axiom,
    ! [N: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,N)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(N),one)) ).

% sub_BitM_One_eq
tff(fact_9173_less__eq__int_Orep__eq,axiom,
    ! [X2: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_aav),aa_int1417691210696471063at_nat(rep_Integ,X2)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_9174_less__int_Orep__eq,axiom,
    ! [X2: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_aat),aa_int1417691210696471063at_nat(rep_Integ,X2)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_9175_nat_Orep__eq,axiom,
    ! [X2: int] : aa_int_nat(nat2,X2) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X2)) ).

% nat.rep_eq
tff(fact_9176_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_aar)) ).

% uminus_int_def
tff(fact_9177_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_9178_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aba,C),set_or4665077453230672383an_nat(X2,Y)) = set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aba,C),set_or4665077453230672383an_nat(X2,Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aba,C),set_or4665077453230672383an_nat(X2,Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9179_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_9180_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9181_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9182_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A2))) ).

% zero_notin_Suc_image
tff(fact_9183_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = set_or1269000886237332187st_nat(one_one_nat,N) ).

% image_Suc_lessThan
tff(fact_9184_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_9185_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9186_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9187_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_9188_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_9189_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: 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,I))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9190_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_aaq)) ).

% times_int_def
tff(fact_9191_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_aaz)) ).

% minus_int_def
tff(fact_9192_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_aax)) ).

% plus_int_def
tff(fact_9193_Gcd__abs__eq,axiom,
    ! [K5: set_int] : gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ).

% Gcd_abs_eq
tff(fact_9194_Gcd__int__eq,axiom,
    ! [N4: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N4)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N4)) ).

% Gcd_int_eq
tff(fact_9195_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_abb,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9196_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_lessThan_int(K2))) ) ).

% finite_int_iff_bounded
tff(fact_9197_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ? [K2: int] : pp(aa_set_int_bool(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_atMost_int(K2))) ) ).

% finite_int_iff_bounded_le
tff(fact_9198_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(A,B)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastAtMost
tff(fact_9199_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,set_or4665077453230672383an_nat(A,B)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastLessThan
tff(fact_9200_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_abc(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9201_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9202_infinite__UNIV__nat,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% infinite_UNIV_nat
tff(fact_9203_nat__not__finite,axiom,
    ~ pp(aa_set_nat_bool(finite_finite_nat,top_top_set_nat)) ).

% nat_not_finite
tff(fact_9204_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_9205_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_9206_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_9207_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_abd,N),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,N) ) ) ).

% range_mod
tff(fact_9208_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9209_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_9210_infinite__UNIV__int,axiom,
    ~ pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ).

% infinite_UNIV_int
tff(fact_9211_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_9212_root__def,axiom,
    ! [N: nat,X2: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X2) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_abe,N)),X2) ) ) ) ).

% root_def
tff(fact_9213_DERIV__even__real__root,axiom,
    ! [N: nat,X2: 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,X2),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(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9214_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X2: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2),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,X2),H3))),aa_real_real(F,X2))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9215_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X2: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2),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,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9216_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X2: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2),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,X2),H3))),aa_real_real(F,X2))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9217_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X2: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2),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,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9218_DERIV__isconst3,axiom,
    ! [A: real,B: real,X2: 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(X2),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,X2) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9219_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))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),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_9220_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))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) ) )
       => 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_9221_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),H3)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9222_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2),H3))),aa_real_real(F,X2))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9223_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2),H3))),aa_real_real(F,X2))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9224_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X2: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,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,X2)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),H3)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9225_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_9226_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X2: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y2: 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,X2),Y2))),D))
             => ( aa_real_real(F,X2) = aa_real_real(F,Y2) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9227_DERIV__ln,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9228_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X2: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y2: 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,X2),Y2))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y2)),aa_real_real(F,X2))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9229_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X2: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y2: 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,X2),Y2))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X2)),aa_real_real(F,Y2))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9230_DERIV__ln__divide,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9231_DERIV__pow,axiom,
    ! [N: nat,X2: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_abf,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(power_power_real(X2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X2,S)) ).

% DERIV_pow
tff(fact_9232_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_abg,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_9233_DERIV__log,axiom,
    ! [X2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => 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)),X2)),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9234_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X2: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X2)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_abh(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,X2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9235_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X2: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X2)))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X2,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_abi(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X2)),aa_real_real(F,X2))),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,X2)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,X2))),aa_real_real(G,X2)))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9236_arcosh__real__has__field__derivative,axiom,
    ! [X2: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => 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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X2,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9237_DERIV__real__sqrt,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9238_DERIV__real__sqrt__generic,axiom,
    ! [X2: real,D5: real] :
      ( ( X2 != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
         => ( D5 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D5,topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9239_artanh__real__has__field__derivative,axiom,
    ! [X2: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X2,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9240_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_abj(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_abl(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_abj(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9241_DERIV__real__root,axiom,
    ! [N: nat,X2: 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),X2))
       => 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(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9242_DERIV__arcsin,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9243_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X2: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M3: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X2)))
          & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_abm(Diff),X2)),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9244_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X2: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X2)))
            & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_abm(Diff),X2)),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9245_DERIV__odd__real__root,axiom,
    ! [N: nat,X2: 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))
     => ( ( X2 != 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(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9246_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 )
         => ( ! [M3: nat,T6: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),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(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),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_abn(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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9247_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 )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),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(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ? [T6: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),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_abn(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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9248_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 )
         => ( ! [M3: nat,T6: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),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_abn(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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9249_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X2: 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))
       => ( ( X2 != zero_zero_real )
         => ( ! [M3: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T6: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X2)))
                & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_abm(Diff),X2)),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9250_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X2: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,T6: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X2))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X2)))
            & ( aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_abm(Diff),X2)),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X2),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9251_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: 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 )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,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_eq_real,C),B))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C))
                  & ( 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_abo(Diff),A),C)),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9252_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: 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 )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
             => ? [T6: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),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_abo(Diff),B),C)),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9253_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X2: 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 )
       => ( ! [M3: nat,T6: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_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))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
                 => ( ( X2 != C )
                   => ? [T6: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T6))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X2)) ) )
                        & ( aa_real_real(F,X2) = 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_abp(Diff),C),X2)),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),T6)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9254_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M3: nat,T6: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),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(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M2: nat,T7: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T7))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T7),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_abr(N),Diff),B2),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T7)),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_abs(Diff),M2),T7)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(T7),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T7,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9255_DERIV__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_abt,suminf_real(aTP_Lamm_abu(X2)),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9256_DERIV__real__root__generic,axiom,
    ! [N: nat,X2: real,D5: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X2 != 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),X2))
             => ( 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(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),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,X2),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(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),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(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N),X2)),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(X2,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9257_DERIV__arccos,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9258_arccos__less__arccos,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2))) ) ) ) ).

% arccos_less_arccos
tff(fact_9259_arccos__less__mono,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),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,X2)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X2)) ) ) ) ).

% arccos_less_mono
tff(fact_9260_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_9261_sin__arccos__nonzero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X2)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9262_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_abv,M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9263_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_bi,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9264_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( 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))
            & ! [X: real] :
                ( ( ( X != C )
                  & 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,C),X))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9265_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X: real] :
                ( ( ( X != C )
                  & 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,C),X))),R3)) )
               => ( aa_real_real(F,X) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9266_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( 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))
            & ! [X: real] :
                ( ( ( X != C )
                  & 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,C),X))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9267_isCont__inverse__function2,axiom,
    ! [A: real,X2: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),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,X2),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9268_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_9269_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_9270_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_abw(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9271_isCont__arcosh,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9272_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D5: real,G: fun_real_real,X2: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D5,topolo2177554685111907308n_real(aa_real_real(G,X2),top_top_set_real))
     => ( ( D5 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
           => ( ! [Y2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),B))
                   => ( aa_real_real(F,aa_real_real(G,Y2)) = Y2 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D5),topolo2177554685111907308n_real(X2,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9273_isCont__arccos,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9274_isCont__arcsin,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9275_LIM__less__bound,axiom,
    ! [B: real,X2: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X2))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(B,X2)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) ) ) ) ).

% LIM_less_bound
tff(fact_9276_isCont__artanh,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9277_isCont__inverse__function,axiom,
    ! [D: real,X2: 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),X2))),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),X2))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X2),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9278_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)) ) )
             => ? [C3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),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,C3)) = 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,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9279_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_abx(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_abx(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_abx(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_9280_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_abx(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_abx(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_abx(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_9281_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_aby(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_abx(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9282_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9283_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ba,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9284_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abz,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9285_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9286_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X6,N2))) )
     => filterlim_nat_real(aTP_Lamm_aca(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9287_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => filterlim_nat_real(aTP_Lamm_acb(C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9288_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),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_9289_LIMSEQ__realpow__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),one_one_real))
       => filterlim_nat_real(power_power_real(X2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9290_LIMSEQ__divide__realpow__zero,axiom,
    ! [X2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_acc(X2),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9291_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(power_power_real(aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_9292_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(power_power_real(C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9293_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X2))
     => filterlim_nat_real(aTP_Lamm_acd(X2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9294_zeroseq__arctan__series,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X2)),one_one_real))
     => filterlim_nat_real(aTP_Lamm_zv(X2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9295_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( 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_abx(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_ace(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_abx(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_aby(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9296_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_aby(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_abx(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9297_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_abx(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_abx(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_9298_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))
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),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_9299_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_acf(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9300_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_acf(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9301_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))
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) )
     => ( 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_9302_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,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),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_9303_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_acg(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9304_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
         => pp(aa_nat_bool(P,N3)) ) ) ).

% eventually_sequentially
tff(fact_9305_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X4))
         => pp(aa_nat_bool(P,X4)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9306_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F3) ) ).

% le_sequentially
tff(fact_9307_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_acg(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9308_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_ach(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9309_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_9310_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))
                & ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z2),X) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9311_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_9312_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_9313_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))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),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_9314_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))
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) ) )
       => ( 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_9315_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X2: 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),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => ( aa_real_real(F,X2) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9316_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_9317_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,set_or6659071591806873216st_nat(L,U))) ).

% finite_greaterThanAtMost
tff(fact_9318_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or6659071591806873216st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9319_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9320_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B: nat] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_9321_Greatest__le__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_9322_GreatestI__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y2: nat] :
            ( pp(aa_nat_bool(P,Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_9323_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: 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),I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9324_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or6656581121297822940st_int(L,U))) ).

% finite_greaterThanAtMost_int
tff(fact_9325_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_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_9326_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9327_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_9328_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_ach(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9329_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9330_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_9331_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9332_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_9333_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_9334_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9335_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_9336_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_acj,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9337_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_acl,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9338_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N9: nat] :
        ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),M4))
         => ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
             => pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N3),M4))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9339_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9340_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_9341_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9342_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_9343_mono__ge2__power__minus__self,axiom,
    ! [K: 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))),K))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acm,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9344_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_9345_Gcd__int__def,axiom,
    ! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Gcd_int_def
tff(fact_9346_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_abe,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_9347_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_acn) ).

% prod_encode_def
tff(fact_9348_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_9349_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_9350_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_9351_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_9352_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_9353_inj__Suc,axiom,
    ! [N4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N4)) ).

% inj_Suc
tff(fact_9354_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aba,K)),N4)) ) ).

% inj_on_diff_nat
tff(fact_9355_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_9356_powr__real__of__int_H,axiom,
    ! [X2: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
     => ( ( ( X2 != 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,X2),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(power_int_real(X2),N) ) ) ) ).

% powr_real_of_int'
tff(fact_9357_pow_Osimps_I1_J,axiom,
    ! [X2: num] : aa_num_num(pow(X2),one) = X2 ).

% pow.simps(1)
tff(fact_9358_pow_Osimps_I3_J,axiom,
    ! [X2: num,Y: num] : aa_num_num(pow(X2),aa_num_num(bit1,Y)) = aa_num_num(times_times_num(aa_num_num(sqr,aa_num_num(pow(X2),Y))),X2) ).

% pow.simps(3)
tff(fact_9359_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X2),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_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))))) )
                    & 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_aco(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9360_sqr_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,N))) ).

% sqr.simps(2)
tff(fact_9361_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9362_sqr__conv__mult,axiom,
    ! [X2: num] : aa_num_num(sqr,X2) = aa_num_num(times_times_num(X2),X2) ).

% sqr_conv_mult
tff(fact_9363_pow_Osimps_I2_J,axiom,
    ! [X2: num,Y: num] : aa_num_num(pow(X2),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(pow(X2),Y)) ).

% pow.simps(2)
tff(fact_9364_sqr_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(sqr,N)),N))) ).

% sqr.simps(3)
tff(fact_9365_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima2,Deg,TreeList),Summary)),Deg3))
    <=> ( ( Deg = Deg3 )
        & ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X3),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_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))))) )
        & 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_aco(Deg),TreeList),Summary)),Mima2) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9366_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X2),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_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))))) )
                & 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_aco(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9367_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X2),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X2 = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X),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(aa_VEB4832260192069861672t_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))))) )
                  & 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_aco(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9368_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_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))))) )
                    & 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_aco(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9369_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X2),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X),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(aa_VEB4832260192069861672t_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))))) )
                      & 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_aco(Deg2),TreeList2),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9370_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,X2),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X2,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X2 = 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] :
                ( ( X2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_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(aa_VEB4832260192069861672t_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))))) )
                      & 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_aco(Deg2),TreeList2),Summary2)),Mima) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9371_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9372_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9373_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9374_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_acp) ).

% Rats_eq_int_div_nat
tff(fact_9375_Rats__dense__in__real,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
     => ? [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_9376_Rats__no__bot__less,axiom,
    ! [X2: 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),X2)) ) ).

% Rats_no_bot_less
tff(fact_9377_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_9378_less__rat,axiom,
    ! [B: int,D: int,A: int,C: 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(C),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,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

% less_rat
tff(fact_9379_Rat__induct__pos,axiom,
    ! [P: 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(P,aa_int_rat(fract(A3),B3))) )
     => pp(aa_rat_bool(P,Q2)) ) ).

% Rat_induct_pos
tff(fact_9380_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_9381_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_9382_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_9383_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_9384_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_9385_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_9386_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_9387_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_9388_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_9389_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9390_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9391_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_9392_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_9393_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_aat)) ).

% less_int_def
tff(fact_9394_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_aav)) ).

% less_eq_int_def
tff(fact_9395_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9396_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = 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,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_9397_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_9398_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9399_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_9400_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_9401_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_9402_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( 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,M),N)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9403_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_9404_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_9405_num__induct,axiom,
    ! [P: fun_num_bool,X2: num] :
      ( pp(aa_num_bool(P,one))
     => ( ! [X4: num] :
            ( pp(aa_num_bool(P,X4))
           => pp(aa_num_bool(P,aa_num_num(inc,X4))) )
       => pp(aa_num_bool(P,X2)) ) ) ).

% num_induct
tff(fact_9406_add__inc,axiom,
    ! [X2: num,Y: num] : aa_num_num(plus_plus_num(X2),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(plus_plus_num(X2),Y)) ).

% add_inc
tff(fact_9407_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9408_inc_Osimps_I2_J,axiom,
    ! [X2: num] : aa_num_num(inc,aa_num_num(bit0,X2)) = aa_num_num(bit1,X2) ).

% inc.simps(2)
tff(fact_9409_inc_Osimps_I3_J,axiom,
    ! [X2: num] : aa_num_num(inc,aa_num_num(bit1,X2)) = aa_num_num(bit0,aa_num_num(inc,X2)) ).

% inc.simps(3)
tff(fact_9410_add__One,axiom,
    ! [X2: num] : aa_num_num(plus_plus_num(X2),one) = aa_num_num(inc,X2) ).

% add_One
tff(fact_9411_inc__BitM__eq,axiom,
    ! [N: num] : aa_num_num(inc,aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_9412_BitM__inc__eq,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(inc,N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_9413_mult__inc,axiom,
    ! [X2: num,Y: num] : aa_num_num(times_times_num(X2),aa_num_num(inc,Y)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(X2),Y)),X2) ).

% mult_inc
tff(fact_9414_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_9415_less__rat__def,axiom,
    ! [X2: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Y))
    <=> pp(aa_rat_bool(positive,aa_rat_rat(minus_minus_rat(Y),X2))) ) ).

% less_rat_def
tff(fact_9416_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_9417_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_9418_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_9419_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_9420_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9421_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),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,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_min_right
tff(fact_9422_nat__mult__min__left,axiom,
    ! [M: 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,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_9423_min__diff,axiom,
    ! [M: nat,I: 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,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_9424_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_acq,N)),M) ).

% min_Suc2
tff(fact_9425_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_acr,N)),M) ).

% min_Suc1
tff(fact_9426_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9427_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_9428_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_9429_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9430_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9431_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9432_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9433_list__encode_Oelims,axiom,
    ! [X2: list_nat,Y: nat] :
      ( ( nat_list_encode(X2) = Y )
     => ( ( ( X2 = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X2 = cons_nat(X4,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9434_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9435_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_9436_upto_Opelims,axiom,
    ! [X2: int,Xa2: int,Y: list_int] :
      ( ( upto(X2,Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X2,Xa2))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa2))
               => ( Y = cons_int(X2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int),Xa2)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa2))
               => ( Y = nil_int ) ) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X2,Xa2)) ) ) ) ).

% upto.pelims
tff(fact_9437_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I,J))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_9438_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9439_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil2
tff(fact_9440_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil
tff(fact_9441_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9442_nth__upto,axiom,
    ! [I: int,K: 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,I),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9443_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9444_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),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,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_9445_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),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,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_9446_upto__rec__numeral_I3_J,axiom,
    ! [M: 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,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),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,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_9447_upto__rec__numeral_I4_J,axiom,
    ! [M: 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,M))),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,M)),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,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),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,M))),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,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_9448_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_9449_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_9450_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9451_distinct__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(distinct_int,upto(I,J))) ).

% distinct_upto
tff(fact_9452_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9453_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9454_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9455_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9456_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9457_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_9458_upto_Oelims,axiom,
    ! [X2: int,Xa2: int,Y: list_int] :
      ( ( upto(X2,Xa2) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa2))
         => ( Y = cons_int(X2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int),Xa2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa2))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_9459_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9460_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9461_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9462_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_9463_cis__Arg__unique,axiom,
    ! [Z: complex,X2: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
         => ( aa_complex_real(arg,Z) = X2 ) ) ) ) ).

% cis_Arg_unique
tff(fact_9464_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_acs(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N)))) ) ).

% bij_betw_roots_unity
tff(fact_9465_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != 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,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_aak(C),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9466_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_act(Z)) ) ) ) ).

% Arg_def
tff(fact_9467_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_9468_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_uf(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_acu(M)) ).

% pairs_le_eq_Sigma
tff(fact_9469_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_9470_list__encode_Opelims,axiom,
    ! [X2: list_nat,Y: nat] :
      ( ( nat_list_encode(X2) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X2))
       => ( ( ( X2 = 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] :
                ( ( X2 = cons_nat(X4,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),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_9471_Restr__natLeq,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,N)),aTP_Lamm_acv(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_acw(N))) ).

% Restr_natLeq
tff(fact_9472_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),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,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_9473_remdups__upt,axiom,
    ! [M: nat,N: nat] : remdups_nat(upt(M,N)) = upt(M,N) ).

% remdups_upt
tff(fact_9474_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_9475_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9476_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).

% drop_upt
tff(fact_9477_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9478_take__upt,axiom,
    ! [I: nat,M: 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,I),M)),N))
     => ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_9479_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9480_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(M,N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_9481_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_9482_nth__upt,axiom,
    ! [I: nat,K: 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,I),K)),J))
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9483_distinct__upt,axiom,
    ! [I: nat,J: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I,J))) ).

% distinct_upt
tff(fact_9484_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : set_or5834768355832116004an_nat(N,M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_9485_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_9486_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or4665077453230672383an_nat(I,J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9487_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or1269000886237332187st_nat(N,M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9488_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or6659071591806873216st_nat(N,M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_9489_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q2) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9490_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9491_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_9492_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9493_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9494_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9495_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X2: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X2,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X2 )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9496_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_9497_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_9498_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9499_Restr__natLeq2,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,N),aTP_Lamm_acx(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_acw(N))) ).

% Restr_natLeq2
tff(fact_9500_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,N)) ).

% natLeq_underS_less
tff(fact_9501_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% not_negative_int_iff
tff(fact_9502_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_9503_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(suc,upt(M,N)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_9504_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acy,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_9505_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_acz,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_9506_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups4561878855575611511st_nat(upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cg),set_or4665077453230672383an_nat(M,N)) ) ) ).

% sum_list_upt
tff(fact_9507_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ada(M),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_adb(M),N4)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_adc(M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9508_card__length__sum__list,axiom,
    ! [M: nat,N4: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ada(M),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),M)),one_one_nat)),N4) ).

% card_length_sum_list
tff(fact_9509_sorted__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).

% sorted_upt
tff(fact_9510_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).

% sorted_wrt_upt
tff(fact_9511_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: 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,I),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9512_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I,J))) ).

% sorted_wrt_upto
tff(fact_9513_sorted__upto,axiom,
    ! [M: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M,N))) ).

% sorted_upto
tff(fact_9514_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_cg,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_9515_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_ch,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_9516_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_acw(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,N)) ).

% Field_natLeq_on
tff(fact_9517_wf__less,axiom,
    wf_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat))) ).

% wf_less
tff(fact_9518_wf__int__ge__less__than2,axiom,
    ! [D: int] : wf_int(int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_9519_wf__int__ge__less__than,axiom,
    ! [D: int] : wf_int(int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_9520_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))
         => ? [K2: nat] :
            ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M4))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),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,M4)),aa_nat_rat(X6,N3)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_9521_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_9522_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))
       => ? [K3: nat] :
          ! [M2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M2))
           => ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),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,M2)),aa_nat_rat(X6,N5)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_9523_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] :
            ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),M3))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),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)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_9524_le__Real,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real2(X6)),real2(Y5)))
        <=> ! [R5: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
             => ? [K2: nat] :
                ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),aa_rat_rat(plus_plus_rat(aa_nat_rat(Y5,N3)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_9525_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))
            & ? [K3: nat] :
              ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),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_9526_vanishes__mult__bounded,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( ? [A8: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A8))
          & ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N2))),A8)) )
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_add(X6),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_9527_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))
         => ? [K2: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),R5)) ) ) ) ).

% vanishes_def
tff(fact_9528_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] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N2))),R3)) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_9529_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))
       => ? [K3: nat] :
          ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),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_9530_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))
            & ? [K3: nat] :
                ( ! [N5: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),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,K3),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_9531_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))
           => ? [K2: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X6,N3)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_9532_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))
            & ? [K2: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X6,N3))) ) ) ) ) ).

% positive_Real
tff(fact_9533_less__real__def,axiom,
    ! [X2: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Y))
    <=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X2))) ) ).

% less_real_def
tff(fact_9534_Real_Opositive_Orep__eq,axiom,
    ! [X2: real] :
      ( pp(aa_real_bool(positive2,X2))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K2: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X2),N3))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9535_Real_Opositive__def,axiom,
    positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_ade) ).

% Real.positive_def
tff(fact_9536_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_9537_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_9538_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))
       => ? [N2: nat] : aa_nat_nat(infini8530281810654367211te_nat(S2),N2) = S ) ) ).

% enumerate_Ex
tff(fact_9539_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_9540_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_9541_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_9542_Least__eq__0,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9543_Least__Suc,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_adf(P))) ) ) ) ).

% Least_Suc
tff(fact_9544_Least__Suc2,axiom,
    ! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(Q,M))
       => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
         => ( ! [K3: nat] :
                ( pp(aa_nat_bool(P,aa_nat_nat(suc,K3)))
              <=> pp(aa_nat_bool(Q,K3)) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9545_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( ( ( K = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9546_less__integer__code_I1_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_integer_code(1)
tff(fact_9547_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = K ) ) ) ).

% abs_integer_code
tff(fact_9548_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X2: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X2)) ) ).

% less_integer.abs_eq
tff(fact_9549_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( sgn_sgn_Code_integer(K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( sgn_sgn_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( sgn_sgn_Code_integer(K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_9550_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9551_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( code_int_of_integer(K) = aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = produc1553301316500091796er_int(aTP_Lamm_adg,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_9552_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = produc1555791787009142072er_nat(aTP_Lamm_adh,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_9553_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9554_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_iff
tff(fact_9555_less__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(X2)),code_int_of_integer(Xa2))) ) ).

% less_integer.rep_eq
tff(fact_9556_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9557_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_adi(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_9558_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9559_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_adj(L),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( ( L = zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
              & ( ( L != zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc6499014454317279255nteger(uminus1351360451143612070nteger,if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_adk(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_9560_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_9561_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_9562_rat__less__code,axiom,
    ! [P3: rat,Q2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q2))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_adm(Q2)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_9563_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q2: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).

% quotient_of_denom_pos
tff(fact_9564_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_on,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_9565_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_acz,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_9566_ATP_Olambda__3,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_mw,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_3
tff(fact_9567_ATP_Olambda__4,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_mz,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_4
tff(fact_9568_ATP_Olambda__5,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_mx,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_5
tff(fact_9569_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_na,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_6
tff(fact_9570_ATP_Olambda__7,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_my,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_7
tff(fact_9571_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_abt,Uu) = suminf_real(aTP_Lamm_yi(Uu)) ).

% ATP.lambda_8
tff(fact_9572_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_abv,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bi,Uu)) ).

% ATP.lambda_9
tff(fact_9573_ATP_Olambda__10,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_acl,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_ack(Uu)))) ).

% ATP.lambda_10
tff(fact_9574_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_acj,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_aci(Uu)))) ).

% ATP.lambda_11
tff(fact_9575_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ff,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_12
tff(fact_9576_ATP_Olambda__13,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_abb,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_13
tff(fact_9577_ATP_Olambda__14,axiom,
    ! [Uu: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_ade,Uu))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K2: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu,N3))) ) ) ) ).

% ATP.lambda_14
tff(fact_9578_ATP_Olambda__15,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_acp,Uu))
    <=> ? [I5: int,N3: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I5)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
          & ( N3 != zero_zero_nat ) ) ) ).

% ATP.lambda_15
tff(fact_9579_ATP_Olambda__16,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xj(Uu),Uua) = if_Code_integer(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_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_16
tff(fact_9580_ATP_Olambda__17,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xh(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_17
tff(fact_9581_ATP_Olambda__18,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xl,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_18
tff(fact_9582_ATP_Olambda__19,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xi(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_19
tff(fact_9583_ATP_Olambda__20,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xk,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_20
tff(fact_9584_ATP_Olambda__21,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_adg,Uu),Uua) = aa_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),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))),code_int_of_integer(Uu))),one_one_int)) ).

% ATP.lambda_21
tff(fact_9585_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_ml,Uu),Uua) = if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),one_one_Code_integer)) ).

% ATP.lambda_22
tff(fact_9586_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_mk,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_23
tff(fact_9587_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mi,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_24
tff(fact_9588_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mn,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_25
tff(fact_9589_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mj,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_26
tff(fact_9590_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mo,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_27
tff(fact_9591_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mm,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_28
tff(fact_9592_ATP_Olambda__29,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_adh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one_nat)) ).

% ATP.lambda_29
tff(fact_9593_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xe(Uu),Uua) = if_Code_integer(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_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_30
tff(fact_9594_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xc(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_31
tff(fact_9595_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xg,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_32
tff(fact_9596_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xd(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_33
tff(fact_9597_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xf,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_34
tff(fact_9598_ATP_Olambda__35,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uj(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ui(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_35
tff(fact_9599_ATP_Olambda__36,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_uh(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ug(Uu),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_36
tff(fact_9600_ATP_Olambda__37,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vi(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vh(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_37
tff(fact_9601_ATP_Olambda__38,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_vg(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ui(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_38
tff(fact_9602_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_mp(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_39
tff(fact_9603_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aab(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(power_power_real(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_40
tff(fact_9604_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xt(Uu),Uua) = times_3573771949741848930nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_41
tff(fact_9605_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xr(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_42
tff(fact_9606_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_43
tff(fact_9607_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xs(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_44
tff(fact_9608_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xu,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_45
tff(fact_9609_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_zq(Uu),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_46
tff(fact_9610_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_zr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_47
tff(fact_9611_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aae,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_48
tff(fact_9612_ATP_Olambda__49,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_act(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_49
tff(fact_9613_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_aaf,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_50
tff(fact_9614_ATP_Olambda__51,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_aag(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_51
tff(fact_9615_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yi(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(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_52
tff(fact_9616_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_abu(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_53
tff(fact_9617_ATP_Olambda__54,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_abx(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_54
tff(fact_9618_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xo(Uu),Uua) = times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_55
tff(fact_9619_ATP_Olambda__56,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_56
tff(fact_9620_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_57
tff(fact_9621_ATP_Olambda__58,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_58
tff(fact_9622_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xp,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_59
tff(fact_9623_ATP_Olambda__60,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ys(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_60
tff(fact_9624_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yu(Uu),Uua) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_61
tff(fact_9625_ATP_Olambda__62,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_yt(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_62
tff(fact_9626_ATP_Olambda__63,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_yj(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_63
tff(fact_9627_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yl(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_64
tff(fact_9628_ATP_Olambda__65,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_yk(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_65
tff(fact_9629_ATP_Olambda__66,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aaa(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_66
tff(fact_9630_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acm,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_67
tff(fact_9631_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_68
tff(fact_9632_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xw,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_69
tff(fact_9633_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_cw,Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_70
tff(fact_9634_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_cv(Uu),Uua))
    <=> ( aa_nat_real(power_power_real(Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_71
tff(fact_9635_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_zv(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(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_72
tff(fact_9636_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ta(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_73
tff(fact_9637_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_tb(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_74
tff(fact_9638_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_td(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_75
tff(fact_9639_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_tc(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_76
tff(fact_9640_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lu(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_77
tff(fact_9641_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lr(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_78
tff(fact_9642_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_lt(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_79
tff(fact_9643_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ls(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_80
tff(fact_9644_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lj(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_81
tff(fact_9645_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lh(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_82
tff(fact_9646_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_li(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_83
tff(fact_9647_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gn(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_84
tff(fact_9648_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gp(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_85
tff(fact_9649_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_go(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_86
tff(fact_9650_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_uk(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_87
tff(fact_9651_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lq(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_88
tff(fact_9652_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lo(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_89
tff(fact_9653_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lp(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_90
tff(fact_9654_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lk(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_91
tff(fact_9655_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ln(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_92
tff(fact_9656_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ll(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_93
tff(fact_9657_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lm(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_94
tff(fact_9658_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nk(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_95
tff(fact_9659_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ni(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_96
tff(fact_9660_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nj(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_97
tff(fact_9661_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uy(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_98
tff(fact_9662_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_uv(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_99
tff(fact_9663_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_us(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_100
tff(fact_9664_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_up(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_101
tff(fact_9665_ATP_Olambda__102,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_kt(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_102
tff(fact_9666_ATP_Olambda__103,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_kw(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_103
tff(fact_9667_ATP_Olambda__104,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_kp(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_104
tff(fact_9668_ATP_Olambda__105,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_kr(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_105
tff(fact_9669_ATP_Olambda__106,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ku(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_106
tff(fact_9670_ATP_Olambda__107,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_kx(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_107
tff(fact_9671_ATP_Olambda__108,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kq(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_108
tff(fact_9672_ATP_Olambda__109,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ks(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_109
tff(fact_9673_ATP_Olambda__110,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kv(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_110
tff(fact_9674_ATP_Olambda__111,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ky(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_111
tff(fact_9675_ATP_Olambda__112,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_rz(Uu),Uua))
    <=> ( aa_complex_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_112
tff(fact_9676_ATP_Olambda__113,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_sc(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_113
tff(fact_9677_ATP_Olambda__114,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_sf(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_114
tff(fact_9678_ATP_Olambda__115,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_rx(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_115
tff(fact_9679_ATP_Olambda__116,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_sa(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_116
tff(fact_9680_ATP_Olambda__117,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_sd(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_117
tff(fact_9681_ATP_Olambda__118,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_sg(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_118
tff(fact_9682_ATP_Olambda__119,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ry(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_119
tff(fact_9683_ATP_Olambda__120,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_sb(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_120
tff(fact_9684_ATP_Olambda__121,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_se(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_121
tff(fact_9685_ATP_Olambda__122,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aby(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_abx(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_122
tff(fact_9686_ATP_Olambda__123,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ace(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_abx(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_123
tff(fact_9687_ATP_Olambda__124,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acn,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_124
tff(fact_9688_ATP_Olambda__125,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ou(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_125
tff(fact_9689_ATP_Olambda__126,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ot(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_126
tff(fact_9690_ATP_Olambda__127,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_abe,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_127
tff(fact_9691_ATP_Olambda__128,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aac(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_128
tff(fact_9692_ATP_Olambda__129,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aad(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_129
tff(fact_9693_ATP_Olambda__130,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_zn(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_130
tff(fact_9694_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_zp,Uu),Uua) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_131
tff(fact_9695_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_zo(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_132
tff(fact_9696_ATP_Olambda__133,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gv,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Uua),one_one_Code_integer)) ).

% ATP.lambda_133
tff(fact_9697_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gt,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_134
tff(fact_9698_ATP_Olambda__135,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gu,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_135
tff(fact_9699_ATP_Olambda__136,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ok(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_136
tff(fact_9700_ATP_Olambda__137,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_om(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_137
tff(fact_9701_ATP_Olambda__138,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ol(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_138
tff(fact_9702_ATP_Olambda__139,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_st(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_139
tff(fact_9703_ATP_Olambda__140,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ss(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_140
tff(fact_9704_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_141
tff(fact_9705_ATP_Olambda__142,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_su(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_142
tff(fact_9706_ATP_Olambda__143,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_tz,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).

% ATP.lambda_143
tff(fact_9707_ATP_Olambda__144,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_au(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_144
tff(fact_9708_ATP_Olambda__145,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_at,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Uua),Uu)) ) ).

% ATP.lambda_145
tff(fact_9709_ATP_Olambda__146,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_av(Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_146
tff(fact_9710_ATP_Olambda__147,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_ua,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_147
tff(fact_9711_ATP_Olambda__148,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_ub,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).

% ATP.lambda_148
tff(fact_9712_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_az,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_9713_ATP_Olambda__150,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_uc,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_150
tff(fact_9714_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abd,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_151
tff(fact_9715_ATP_Olambda__152,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_nf,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_152
tff(fact_9716_ATP_Olambda__153,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_nb,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,Uua),Uu)) ) ).

% ATP.lambda_153
tff(fact_9717_ATP_Olambda__154,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_nc,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_154
tff(fact_9718_ATP_Olambda__155,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_nd,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).

% ATP.lambda_155
tff(fact_9719_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ay,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_156
tff(fact_9720_ATP_Olambda__157,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ne,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_157
tff(fact_9721_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_158
tff(fact_9722_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aba,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_159
tff(fact_9723_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_abf,Uu),Uua) = aa_nat_real(power_power_real(Uua),Uu) ).

% ATP.lambda_160
tff(fact_9724_ATP_Olambda__161,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_161
tff(fact_9725_ATP_Olambda__162,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_abc(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_162
tff(fact_9726_ATP_Olambda__163,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_abg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_163
tff(fact_9727_ATP_Olambda__164,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_bh,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_164
tff(fact_9728_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bi,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_165
tff(fact_9729_ATP_Olambda__166,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ba,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_166
tff(fact_9730_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_aar,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_167
tff(fact_9731_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_acb(Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_168
tff(fact_9732_ATP_Olambda__169,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uua),Uu)) ) ).

% ATP.lambda_169
tff(fact_9733_ATP_Olambda__170,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_170
tff(fact_9734_ATP_Olambda__171,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_171
tff(fact_9735_ATP_Olambda__172,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_172
tff(fact_9736_ATP_Olambda__173,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_173
tff(fact_9737_ATP_Olambda__174,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_174
tff(fact_9738_ATP_Olambda__175,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_fl,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_175
tff(fact_9739_ATP_Olambda__176,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_fj(Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_176
tff(fact_9740_ATP_Olambda__177,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_fk,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_177
tff(fact_9741_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fm,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_178
tff(fact_9742_ATP_Olambda__179,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fn,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_179
tff(fact_9743_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_acx(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_180
tff(fact_9744_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oh(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_181
tff(fact_9745_ATP_Olambda__182,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_og(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_182
tff(fact_9746_ATP_Olambda__183,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rm(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_183
tff(fact_9747_ATP_Olambda__184,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_km(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_184
tff(fact_9748_ATP_Olambda__185,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_adf(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_185
tff(fact_9749_ATP_Olambda__186,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lf(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_186
tff(fact_9750_ATP_Olambda__187,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_kl(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_187
tff(fact_9751_ATP_Olambda__188,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lg(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_188
tff(fact_9752_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aaz,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aay(Uu),Uua)) ).

% ATP.lambda_189
tff(fact_9753_ATP_Olambda__190,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aax,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aaw(Uu),Uua)) ).

% ATP.lambda_190
tff(fact_9754_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_aav,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_aau(Uu),Uua)) ).

% ATP.lambda_191
tff(fact_9755_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_aat,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_aas(Uu),Uua)) ).

% ATP.lambda_192
tff(fact_9756_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aaq,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aap(Uu),Uua)) ).

% ATP.lambda_193
tff(fact_9757_ATP_Olambda__194,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_abl(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_abk(Uu),Uua)) ).

% ATP.lambda_194
tff(fact_9758_ATP_Olambda__195,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_acs(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_195
tff(fact_9759_ATP_Olambda__196,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_zw(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_196
tff(fact_9760_ATP_Olambda__197,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_zy(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_197
tff(fact_9761_ATP_Olambda__198,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_zx(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_198
tff(fact_9762_ATP_Olambda__199,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_oo(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_199
tff(fact_9763_ATP_Olambda__200,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_or(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_200
tff(fact_9764_ATP_Olambda__201,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_op(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_201
tff(fact_9765_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_os(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_202
tff(fact_9766_ATP_Olambda__203,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_oq(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_203
tff(fact_9767_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_acu(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_204
tff(fact_9768_ATP_Olambda__205,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_acd(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_205
tff(fact_9769_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_206
tff(fact_9770_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acq,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_207
tff(fact_9771_ATP_Olambda__208,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_max_nat,Uu),Uua)) ).

% ATP.lambda_208
tff(fact_9772_ATP_Olambda__209,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_max_nat,Uua),Uu)) ).

% ATP.lambda_209
tff(fact_9773_ATP_Olambda__210,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_eo(Uu),Uua))
    <=> ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uua),Uu)) ) ).

% ATP.lambda_210
tff(fact_9774_ATP_Olambda__211,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ep(Uu),Uua))
    <=> ~ pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_211
tff(fact_9775_ATP_Olambda__212,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_eq(Uu),Uua))
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_212
tff(fact_9776_ATP_Olambda__213,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_er(Uu),Uua))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_213
tff(fact_9777_ATP_Olambda__214,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_es(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_214
tff(fact_9778_ATP_Olambda__215,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_et(Uu),Uua))
    <=> ~ pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_215
tff(fact_9779_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_uo(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_9780_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aca(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_9781_ATP_Olambda__218,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_lx(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_9782_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_ma(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_9783_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_lz(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_9784_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_ly(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_9785_ATP_Olambda__222,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_lw(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_9786_ATP_Olambda__223,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_va(Uu),Uua) = aa_real_real(ln_ln_real,aa_VEBT_VEBT_real(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_9787_ATP_Olambda__224,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_ve(Uu),Uua) = aa_real_real(ln_ln_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_9788_ATP_Olambda__225,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_vc(Uu),Uua) = aa_real_real(ln_ln_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_9789_ATP_Olambda__226,axiom,
    ! [Uu: fun_set_nat_real,Uua: set_nat] : aa_set_nat_real(aTP_Lamm_vb(Uu),Uua) = aa_real_real(ln_ln_real,aa_set_nat_real(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_9790_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vf(Uu),Uua) = aa_real_real(ln_ln_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_9791_ATP_Olambda__228,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_vd(Uu),Uua) = aa_real_real(ln_ln_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_9792_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_qs(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_9793_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_qt(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_9794_ATP_Olambda__231,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_qu(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_9795_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_qw(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_9796_ATP_Olambda__233,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_qv(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_233
tff(fact_9797_ATP_Olambda__234,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_gi(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_234
tff(fact_9798_ATP_Olambda__235,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_gj(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_235
tff(fact_9799_ATP_Olambda__236,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_236
tff(fact_9800_ATP_Olambda__237,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_237
tff(fact_9801_ATP_Olambda__238,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_gk(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_238
tff(fact_9802_ATP_Olambda__239,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_gl(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_239
tff(fact_9803_ATP_Olambda__240,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_gm(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_240
tff(fact_9804_ATP_Olambda__241,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_te(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_241
tff(fact_9805_ATP_Olambda__242,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ti(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_242
tff(fact_9806_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_tf(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_243
tff(fact_9807_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_th(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_244
tff(fact_9808_ATP_Olambda__245,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_tg(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_245
tff(fact_9809_ATP_Olambda__246,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yr(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_246
tff(fact_9810_ATP_Olambda__247,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_yq(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_247
tff(fact_9811_ATP_Olambda__248,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_fa(Uu),Uua))
    <=> ~ pp(aa_real_bool(Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_9812_ATP_Olambda__249,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_fb(Uu),Uua))
    <=> ~ pp(aa_list_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_9813_ATP_Olambda__250,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_fc(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_9814_ATP_Olambda__251,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_fd(Uu),Uua))
    <=> ~ pp(aa_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_9815_ATP_Olambda__252,axiom,
    ! [Uu: fun_int_bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_fe(Uu),Uua))
    <=> ~ pp(aa_int_bool(Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_9816_ATP_Olambda__253,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_acv(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay,Uu)) ).

% ATP.lambda_253
tff(fact_9817_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_of(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_254
tff(fact_9818_ATP_Olambda__255,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_el(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Uua),Uub)) ).

% ATP.lambda_255
tff(fact_9819_ATP_Olambda__256,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_en(Uu),Uua),Uub) = 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_256
tff(fact_9820_ATP_Olambda__257,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_em(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_257
tff(fact_9821_ATP_Olambda__258,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rt(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_258
tff(fact_9822_ATP_Olambda__259,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rw(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_259
tff(fact_9823_ATP_Olambda__260,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rv(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_260
tff(fact_9824_ATP_Olambda__261,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ru(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_261
tff(fact_9825_ATP_Olambda__262,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_fo(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_262
tff(fact_9826_ATP_Olambda__263,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_fs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_263
tff(fact_9827_ATP_Olambda__264,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_fw(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub),aa_VEBT_VEBT_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_264
tff(fact_9828_ATP_Olambda__265,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fv(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_265
tff(fact_9829_ATP_Olambda__266,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_fp(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_266
tff(fact_9830_ATP_Olambda__267,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ft(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_267
tff(fact_9831_ATP_Olambda__268,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fx(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_268
tff(fact_9832_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fq(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_269
tff(fact_9833_ATP_Olambda__270,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_fr(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_270
tff(fact_9834_ATP_Olambda__271,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fu(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_271
tff(fact_9835_ATP_Olambda__272,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_ox(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub),aa_VEBT_VEBT_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_272
tff(fact_9836_ATP_Olambda__273,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_pc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub)),aa_VEBT_VEBT_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_273
tff(fact_9837_ATP_Olambda__274,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_pb(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_274
tff(fact_9838_ATP_Olambda__275,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_pg(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_275
tff(fact_9839_ATP_Olambda__276,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_oy(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_276
tff(fact_9840_ATP_Olambda__277,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pd(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_277
tff(fact_9841_ATP_Olambda__278,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oz(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)),one_one_complex) ).

% ATP.lambda_278
tff(fact_9842_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pe(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_279
tff(fact_9843_ATP_Olambda__280,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_pa(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)),one_one_complex) ).

% ATP.lambda_280
tff(fact_9844_ATP_Olambda__281,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_pf(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_281
tff(fact_9845_ATP_Olambda__282,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_fy(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_282
tff(fact_9846_ATP_Olambda__283,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_gc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_283
tff(fact_9847_ATP_Olambda__284,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_gg(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu),aa_VEBT_VEBT_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_284
tff(fact_9848_ATP_Olambda__285,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_gf(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_285
tff(fact_9849_ATP_Olambda__286,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_fz(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_286
tff(fact_9850_ATP_Olambda__287,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gd(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_287
tff(fact_9851_ATP_Olambda__288,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gh(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_288
tff(fact_9852_ATP_Olambda__289,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ga(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_289
tff(fact_9853_ATP_Olambda__290,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pr(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_290
tff(fact_9854_ATP_Olambda__291,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ps(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_291
tff(fact_9855_ATP_Olambda__292,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pt(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_292
tff(fact_9856_ATP_Olambda__293,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_gb(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_293
tff(fact_9857_ATP_Olambda__294,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ge(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_294
tff(fact_9858_ATP_Olambda__295,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_ph(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu),aa_VEBT_VEBT_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_295
tff(fact_9859_ATP_Olambda__296,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_pm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu)),aa_VEBT_VEBT_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_296
tff(fact_9860_ATP_Olambda__297,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_pl(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_297
tff(fact_9861_ATP_Olambda__298,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_pq(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_298
tff(fact_9862_ATP_Olambda__299,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_pi(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_299
tff(fact_9863_ATP_Olambda__300,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pn(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_300
tff(fact_9864_ATP_Olambda__301,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pj(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)),one_one_complex) ).

% ATP.lambda_301
tff(fact_9865_ATP_Olambda__302,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_po(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_302
tff(fact_9866_ATP_Olambda__303,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_pk(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)),one_one_complex) ).

% ATP.lambda_303
tff(fact_9867_ATP_Olambda__304,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_pp(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_304
tff(fact_9868_ATP_Olambda__305,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_adk(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_305
tff(fact_9869_ATP_Olambda__306,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_adj(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_306
tff(fact_9870_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rp(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uu,Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_307
tff(fact_9871_ATP_Olambda__308,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_rs(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_308
tff(fact_9872_ATP_Olambda__309,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rr(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_309
tff(fact_9873_ATP_Olambda__310,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_rq(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_310
tff(fact_9874_ATP_Olambda__311,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6792494735270171281omplex(aTP_Lamm_jv(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_311
tff(fact_9875_ATP_Olambda__312,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_jz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_312
tff(fact_9876_ATP_Olambda__313,axiom,
    ! [Uu: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun3193099451372820155BT_rat(aTP_Lamm_ke(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_313
tff(fact_9877_ATP_Olambda__314,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_kd(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_314
tff(fact_9878_ATP_Olambda__315,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_jw(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_315
tff(fact_9879_ATP_Olambda__316,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_ka(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_316
tff(fact_9880_ATP_Olambda__317,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_kf(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_317
tff(fact_9881_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_jx(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_318
tff(fact_9882_ATP_Olambda__319,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_jy(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_319
tff(fact_9883_ATP_Olambda__320,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_kb(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_320
tff(fact_9884_ATP_Olambda__321,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6792494735270171281omplex(aTP_Lamm_rb(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_321
tff(fact_9885_ATP_Olambda__322,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_rg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_322
tff(fact_9886_ATP_Olambda__323,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_bool,Uub: complex] : aa_complex_complex(aa_fun1034237174562290143omplex(aTP_Lamm_rf(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(Uua,Uub),aa_complex_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_323
tff(fact_9887_ATP_Olambda__324,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_rk(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_324
tff(fact_9888_ATP_Olambda__325,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_rc(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_325
tff(fact_9889_ATP_Olambda__326,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_rh(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_326
tff(fact_9890_ATP_Olambda__327,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_rd(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_327
tff(fact_9891_ATP_Olambda__328,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_bool,Uub: nat] : aa_nat_real(aa_fun1750223370215434333t_real(aTP_Lamm_ri(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_328
tff(fact_9892_ATP_Olambda__329,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_re(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_329
tff(fact_9893_ATP_Olambda__330,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_rj(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_330
tff(fact_9894_ATP_Olambda__331,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vh(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_331
tff(fact_9895_ATP_Olambda__332,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ui(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_332
tff(fact_9896_ATP_Olambda__333,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ug(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_333
tff(fact_9897_ATP_Olambda__334,axiom,
    ! [Uu: fun_VE2006399705683203104omplex,Uua: complex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_com2862003112617944841omplex(aTP_Lamm_iv(Uu),Uua),Uub) = aa_complex_complex(aa_VEB5608662635949521omplex(Uu,Uub),Uua) ).

% ATP.lambda_334
tff(fact_9898_ATP_Olambda__335,axiom,
    ! [Uu: fun_VE597728521808591204at_nat,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_ie(Uu),Uua),Uub) = aa_nat_nat(aa_VEB1122092716774696213at_nat(Uu,Uub),Uua) ).

% ATP.lambda_335
tff(fact_9899_ATP_Olambda__336,axiom,
    ! [Uu: fun_VE7331017413731640256at_int,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_pw(Uu),Uua),Uub) = aa_nat_int(aa_VEB6167613734120275313at_int(Uu,Uub),Uua) ).

% ATP.lambda_336
tff(fact_9900_ATP_Olambda__337,axiom,
    ! [Uu: fun_VE294566951569886236nt_int,Uua: int,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_int1343382402272391561BT_int(aTP_Lamm_hn(Uu),Uua),Uub) = aa_int_int(aa_VEB7166967446021329869nt_int(Uu,Uub),Uua) ).

% ATP.lambda_337
tff(fact_9901_ATP_Olambda__338,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_iq(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_338
tff(fact_9902_ATP_Olambda__339,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_qc(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_339
tff(fact_9903_ATP_Olambda__340,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_hz(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_340
tff(fact_9904_ATP_Olambda__341,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_iz(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_341
tff(fact_9905_ATP_Olambda__342,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_ii(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_342
tff(fact_9906_ATP_Olambda__343,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_pz(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_343
tff(fact_9907_ATP_Olambda__344,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_hr(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_344
tff(fact_9908_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ut(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_345
tff(fact_9909_ATP_Olambda__346,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_346
tff(fact_9910_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_uw(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_347
tff(fact_9911_ATP_Olambda__348,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hv(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_348
tff(fact_9912_ATP_Olambda__349,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_im(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_349
tff(fact_9913_ATP_Olambda__350,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sj(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_350
tff(fact_9914_ATP_Olambda__351,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sk(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(power_power_real(Uua),Uub)) ).

% ATP.lambda_351
tff(fact_9915_ATP_Olambda__352,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_sl(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_352
tff(fact_9916_ATP_Olambda__353,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_adi(Uu),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).

% ATP.lambda_353
tff(fact_9917_ATP_Olambda__354,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ux(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_uw(Uu),Uub),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_354
tff(fact_9918_ATP_Olambda__355,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_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uq(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_355
tff(fact_9919_ATP_Olambda__356,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_uu(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ut(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_356
tff(fact_9920_ATP_Olambda__357,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ur(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uq(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_357
tff(fact_9921_ATP_Olambda__358,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_adm(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_adl(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_358
tff(fact_9922_ATP_Olambda__359,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_abm(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(power_power_real(Uua),Uub)) ).

% ATP.lambda_359
tff(fact_9923_ATP_Olambda__360,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_abn(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(power_power_real(Uu),Uub)) ).

% ATP.lambda_360
tff(fact_9924_ATP_Olambda__361,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_vo(Uu),Uua),Uub))
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ts(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_361
tff(fact_9925_ATP_Olambda__362,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_vn(Uu),Uua),Uub))
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_tj(Uu),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_362
tff(fact_9926_ATP_Olambda__363,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gq(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_363
tff(fact_9927_ATP_Olambda__364,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gs(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(power_power_real(zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_364
tff(fact_9928_ATP_Olambda__365,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gr(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_365
tff(fact_9929_ATP_Olambda__366,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_abj(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(power_power_real(Uua),Uub)) ).

% ATP.lambda_366
tff(fact_9930_ATP_Olambda__367,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_zz(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(power_power_real(Uu),Uub)) ).

% ATP.lambda_367
tff(fact_9931_ATP_Olambda__368,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_adb(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_368
tff(fact_9932_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_yn(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_369
tff(fact_9933_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_370
tff(fact_9934_ATP_Olambda__371,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_adc(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_371
tff(fact_9935_ATP_Olambda__372,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cq(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(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_372
tff(fact_9936_ATP_Olambda__373,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_cp(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_373
tff(fact_9937_ATP_Olambda__374,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)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Uub)),Uua)) ) ) ).

% ATP.lambda_374
tff(fact_9938_ATP_Olambda__375,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))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_375
tff(fact_9939_ATP_Olambda__376,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))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Uub)),Uua)) ) ) ).

% ATP.lambda_376
tff(fact_9940_ATP_Olambda__377,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu))
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_377
tff(fact_9941_ATP_Olambda__378,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_ck(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_378
tff(fact_9942_ATP_Olambda__379,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_379
tff(fact_9943_ATP_Olambda__380,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_cn(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_380
tff(fact_9944_ATP_Olambda__381,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_co(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_381
tff(fact_9945_ATP_Olambda__382,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ada(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_382
tff(fact_9946_ATP_Olambda__383,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aai(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_383
tff(fact_9947_ATP_Olambda__384,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_aao(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu))))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_384
tff(fact_9948_ATP_Olambda__385,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_acw(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_385
tff(fact_9949_ATP_Olambda__386,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aaj(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_386
tff(fact_9950_ATP_Olambda__387,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ym(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_387
tff(fact_9951_ATP_Olambda__388,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_388
tff(fact_9952_ATP_Olambda__389,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_zg(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_389
tff(fact_9953_ATP_Olambda__390,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_zf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_390
tff(fact_9954_ATP_Olambda__391,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xy(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_391
tff(fact_9955_ATP_Olambda__392,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_gw(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uua)) ) ) ).

% ATP.lambda_392
tff(fact_9956_ATP_Olambda__393,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_gx(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_393
tff(fact_9957_ATP_Olambda__394,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_gy(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_394
tff(fact_9958_ATP_Olambda__395,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_gz(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_395
tff(fact_9959_ATP_Olambda__396,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ha(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_396
tff(fact_9960_ATP_Olambda__397,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_hb(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_397
tff(fact_9961_ATP_Olambda__398,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fg(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_398
tff(fact_9962_ATP_Olambda__399,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ar(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_399
tff(fact_9963_ATP_Olambda__400,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fh(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_400
tff(fact_9964_ATP_Olambda__401,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ax(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_401
tff(fact_9965_ATP_Olambda__402,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_aw(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_402
tff(fact_9966_ATP_Olambda__403,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_as(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_403
tff(fact_9967_ATP_Olambda__404,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aTP_Lamm_bb(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & pp(aa_VEBT_VEBT_bool(Uua,Uub)) ) ) ).

% ATP.lambda_404
tff(fact_9968_ATP_Olambda__405,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_kc(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_405
tff(fact_9969_ATP_Olambda__406,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_bc(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_406
tff(fact_9970_ATP_Olambda__407,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_bd(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_407
tff(fact_9971_ATP_Olambda__408,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_be(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_408
tff(fact_9972_ATP_Olambda__409,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_bf(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_409
tff(fact_9973_ATP_Olambda__410,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_bg(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_410
tff(fact_9974_ATP_Olambda__411,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jd(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_411
tff(fact_9975_ATP_Olambda__412,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_jb(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_412
tff(fact_9976_ATP_Olambda__413,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_jc(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_413
tff(fact_9977_ATP_Olambda__414,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_je(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_414
tff(fact_9978_ATP_Olambda__415,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jf(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_415
tff(fact_9979_ATP_Olambda__416,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ji(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_416
tff(fact_9980_ATP_Olambda__417,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_jg(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_417
tff(fact_9981_ATP_Olambda__418,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_jh(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_418
tff(fact_9982_ATP_Olambda__419,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_jj(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_419
tff(fact_9983_ATP_Olambda__420,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jk(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_420
tff(fact_9984_ATP_Olambda__421,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aTP_Lamm_dr(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_421
tff(fact_9985_ATP_Olambda__422,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_eb(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_422
tff(fact_9986_ATP_Olambda__423,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_dz(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_423
tff(fact_9987_ATP_Olambda__424,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_ej(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_424
tff(fact_9988_ATP_Olambda__425,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_dt(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_425
tff(fact_9989_ATP_Olambda__426,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_ed(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_426
tff(fact_9990_ATP_Olambda__427,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_dv(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_427
tff(fact_9991_ATP_Olambda__428,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ef(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_428
tff(fact_9992_ATP_Olambda__429,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_dx(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_429
tff(fact_9993_ATP_Olambda__430,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_eh(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_430
tff(fact_9994_ATP_Olambda__431,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aTP_Lamm_cx(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_431
tff(fact_9995_ATP_Olambda__432,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_dh(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_432
tff(fact_9996_ATP_Olambda__433,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_df(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_433
tff(fact_9997_ATP_Olambda__434,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_dp(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_434
tff(fact_9998_ATP_Olambda__435,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_cz(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_435
tff(fact_9999_ATP_Olambda__436,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_dj(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_436
tff(fact_10000_ATP_Olambda__437,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_db(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_437
tff(fact_10001_ATP_Olambda__438,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_dl(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_438
tff(fact_10002_ATP_Olambda__439,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_dd(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_439
tff(fact_10003_ATP_Olambda__440,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_dn(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_440
tff(fact_10004_ATP_Olambda__441,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_eu(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uu))
        & ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uub),Uua)) ) ) ).

% ATP.lambda_441
tff(fact_10005_ATP_Olambda__442,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ev(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ~ pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_442
tff(fact_10006_ATP_Olambda__443,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_ew(Uu),Uua),Uub))
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_443
tff(fact_10007_ATP_Olambda__444,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ex(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_444
tff(fact_10008_ATP_Olambda__445,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ez(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_445
tff(fact_10009_ATP_Olambda__446,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_ey(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_446
tff(fact_10010_ATP_Olambda__447,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abw(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_447
tff(fact_10011_ATP_Olambda__448,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_uf(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_448
tff(fact_10012_ATP_Olambda__449,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_ack(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu)) ) ).

% ATP.lambda_449
tff(fact_10013_ATP_Olambda__450,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aci(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_450
tff(fact_10014_ATP_Olambda__451,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ul(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_451
tff(fact_10015_ATP_Olambda__452,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_cu(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_452
tff(fact_10016_ATP_Olambda__453,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_aak(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_453
tff(fact_10017_ATP_Olambda__454,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_vj(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_454
tff(fact_10018_ATP_Olambda__455,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_455
tff(fact_10019_ATP_Olambda__456,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_abk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_456
tff(fact_10020_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tj(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_457
tff(fact_10021_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ts(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_458
tff(fact_10022_ATP_Olambda__459,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_tm(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_459
tff(fact_10023_ATP_Olambda__460,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vx(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_460
tff(fact_10024_ATP_Olambda__461,axiom,
    ! [Uu: fun_nat_int,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_nat_int(Uu,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_461
tff(fact_10025_ATP_Olambda__462,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ci(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_462
tff(fact_10026_ATP_Olambda__463,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_add(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_463
tff(fact_10027_ATP_Olambda__464,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aah(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_464
tff(fact_10028_ATP_Olambda__465,axiom,
    ! [Uu: fun_VE9219535225822561060nteger,Uua: fun_VE9219535225822561060nteger,Uub: vEBT_VEBT] : aa_VEB5935388676523804373nteger(aa_fun2400605388621188028nteger(aTP_Lamm_yw(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_VEB5935388676523804373nteger(Uua,Uub),aa_VEB5935388676523804373nteger(Uu,Uub)) ).

% ATP.lambda_465
tff(fact_10029_ATP_Olambda__466,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_yz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uub)),aa_VEBT_VEBT_real(Uu,Uub)) ).

% ATP.lambda_466
tff(fact_10030_ATP_Olambda__467,axiom,
    ! [Uu: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_zd(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(Uua,Uub)),aa_VEBT_VEBT_rat(Uu,Uub)) ).

% ATP.lambda_467
tff(fact_10031_ATP_Olambda__468,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_yy(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_real_Code_integer(Uua,Uub),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_468
tff(fact_10032_ATP_Olambda__469,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_469
tff(fact_10033_ATP_Olambda__470,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_yv(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_nat_Code_integer(Uua,Uub),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_470
tff(fact_10034_ATP_Olambda__471,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_zc(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_471
tff(fact_10035_ATP_Olambda__472,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_yx(Uu),Uua),Uub) = times_3573771949741848930nteger(aa_int_Code_integer(Uua,Uub),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_472
tff(fact_10036_ATP_Olambda__473,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_za(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_473
tff(fact_10037_ATP_Olambda__474,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ze(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uub)),aa_int_rat(Uu,Uub)) ).

% ATP.lambda_474
tff(fact_10038_ATP_Olambda__475,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_kg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_VEBT_VEBT_nat(Uua,Uub)),aa_VEBT_VEBT_nat(Uu,Uub)) ).

% ATP.lambda_475
tff(fact_10039_ATP_Olambda__476,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kj(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_476
tff(fact_10040_ATP_Olambda__477,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ki(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_477
tff(fact_10041_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_478
tff(fact_10042_ATP_Olambda__479,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kk(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_479
tff(fact_10043_ATP_Olambda__480,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kh(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_480
tff(fact_10044_ATP_Olambda__481,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_hj(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_481
tff(fact_10045_ATP_Olambda__482,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_482
tff(fact_10046_ATP_Olambda__483,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hi(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_483
tff(fact_10047_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pu(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_484
tff(fact_10048_ATP_Olambda__485,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hh(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_485
tff(fact_10049_ATP_Olambda__486,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_abi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_486
tff(fact_10050_ATP_Olambda__487,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_487
tff(fact_10051_ATP_Olambda__488,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_488
tff(fact_10052_ATP_Olambda__489,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_489
tff(fact_10053_ATP_Olambda__490,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_490
tff(fact_10054_ATP_Olambda__491,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_491
tff(fact_10055_ATP_Olambda__492,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_492
tff(fact_10056_ATP_Olambda__493,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_aq(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_493
tff(fact_10057_ATP_Olambda__494,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_al(Uu),Uua),Uub))
    <=> ( pp(aa_real_bool(Uu,Uub))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_494
tff(fact_10058_ATP_Olambda__495,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_am(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_495
tff(fact_10059_ATP_Olambda__496,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_an(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_496
tff(fact_10060_ATP_Olambda__497,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ao(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_497
tff(fact_10061_ATP_Olambda__498,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ap(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_498
tff(fact_10062_ATP_Olambda__499,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_499
tff(fact_10063_ATP_Olambda__500,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tv(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_500
tff(fact_10064_ATP_Olambda__501,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_501
tff(fact_10065_ATP_Olambda__502,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qz(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_502
tff(fact_10066_ATP_Olambda__503,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ra(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_503
tff(fact_10067_ATP_Olambda__504,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_acf(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_504
tff(fact_10068_ATP_Olambda__505,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_abh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_505
tff(fact_10069_ATP_Olambda__506,axiom,
    ! [Uu: real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_hc(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_506
tff(fact_10070_ATP_Olambda__507,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_hd(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_507
tff(fact_10071_ATP_Olambda__508,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_he(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_508
tff(fact_10072_ATP_Olambda__509,axiom,
    ! [Uu: nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_hf(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_509
tff(fact_10073_ATP_Olambda__510,axiom,
    ! [Uu: int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_hg(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_510
tff(fact_10074_ATP_Olambda__511,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_vm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_511
tff(fact_10075_ATP_Olambda__512,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_me(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Uub),Uua)) ).

% ATP.lambda_512
tff(fact_10076_ATP_Olambda__513,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_md(Uu),Uua),Uub) = plus_p3455044024723400733d_enat(Uu,times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub),Uua)) ).

% ATP.lambda_513
tff(fact_10077_ATP_Olambda__514,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mb(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_514
tff(fact_10078_ATP_Olambda__515,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mh(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_515
tff(fact_10079_ATP_Olambda__516,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_mc(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_516
tff(fact_10080_ATP_Olambda__517,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mg(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_517
tff(fact_10081_ATP_Olambda__518,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_mf(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_518
tff(fact_10082_ATP_Olambda__519,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ob(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_519
tff(fact_10083_ATP_Olambda__520,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_oe(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_520
tff(fact_10084_ATP_Olambda__521,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oc(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_521
tff(fact_10085_ATP_Olambda__522,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_od(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_522
tff(fact_10086_ATP_Olambda__523,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_523
tff(fact_10087_ATP_Olambda__524,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_kz(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_524
tff(fact_10088_ATP_Olambda__525,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lc(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_525
tff(fact_10089_ATP_Olambda__526,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_la(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_526
tff(fact_10090_ATP_Olambda__527,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lb(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_527
tff(fact_10091_ATP_Olambda__528,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lv(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_528
tff(fact_10092_ATP_Olambda__529,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ach(Uu),Uua),Uub))
    <=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua))) ) ).

% ATP.lambda_529
tff(fact_10093_ATP_Olambda__530,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_qx(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_530
tff(fact_10094_ATP_Olambda__531,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_531
tff(fact_10095_ATP_Olambda__532,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sx(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_532
tff(fact_10096_ATP_Olambda__533,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_sw(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_533
tff(fact_10097_ATP_Olambda__534,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sz(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_534
tff(fact_10098_ATP_Olambda__535,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sy(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_535
tff(fact_10099_ATP_Olambda__536,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nh(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_536
tff(fact_10100_ATP_Olambda__537,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ng(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_537
tff(fact_10101_ATP_Olambda__538,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sh(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_538
tff(fact_10102_ATP_Olambda__539,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_vl(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_539
tff(fact_10103_ATP_Olambda__540,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ko(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_540
tff(fact_10104_ATP_Olambda__541,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_acg(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_541
tff(fact_10105_ATP_Olambda__542,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kn(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_542
tff(fact_10106_ATP_Olambda__543,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_rl(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_543
tff(fact_10107_ATP_Olambda__544,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_rn(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_544
tff(fact_10108_ATP_Olambda__545,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ro(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_545
tff(fact_10109_ATP_Olambda__546,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_ue(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_546
tff(fact_10110_ATP_Olambda__547,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ud(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub))) ).

% ATP.lambda_547
tff(fact_10111_ATP_Olambda__548,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wa(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_548
tff(fact_10112_ATP_Olambda__549,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wi(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_549
tff(fact_10113_ATP_Olambda__550,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wc(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_550
tff(fact_10114_ATP_Olambda__551,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wg(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_551
tff(fact_10115_ATP_Olambda__552,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_we(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_552
tff(fact_10116_ATP_Olambda__553,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wk(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_553
tff(fact_10117_ATP_Olambda__554,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wl(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_554
tff(fact_10118_ATP_Olambda__555,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wm(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_555
tff(fact_10119_ATP_Olambda__556,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(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_556
tff(fact_10120_ATP_Olambda__557,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wn(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_557
tff(fact_10121_ATP_Olambda__558,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wb(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_558
tff(fact_10122_ATP_Olambda__559,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wj(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_559
tff(fact_10123_ATP_Olambda__560,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wd(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_560
tff(fact_10124_ATP_Olambda__561,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wh(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_561
tff(fact_10125_ATP_Olambda__562,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wf(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_562
tff(fact_10126_ATP_Olambda__563,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_sn(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_563
tff(fact_10127_ATP_Olambda__564,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_jl(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uuc),Uu)),aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) ).

% ATP.lambda_564
tff(fact_10128_ATP_Olambda__565,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_jn(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uuc),Uu),aa_VEBT_VEBT_rat(Uua,Uuc)),aa_VEBT_VEBT_rat(Uub,Uuc)) ).

% ATP.lambda_565
tff(fact_10129_ATP_Olambda__566,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_jp(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uuc),Uu)),aa_VEBT_VEBT_nat(Uua,Uuc)),aa_VEBT_VEBT_nat(Uub,Uuc)) ).

% ATP.lambda_566
tff(fact_10130_ATP_Olambda__567,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_int,Uub: fun_VEBT_VEBT_int,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_jr(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uuc),Uu),aa_VEBT_VEBT_int(Uua,Uuc)),aa_VEBT_VEBT_int(Uub,Uuc)) ).

% ATP.lambda_567
tff(fact_10131_ATP_Olambda__568,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_jm(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_568
tff(fact_10132_ATP_Olambda__569,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_jo(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_569
tff(fact_10133_ATP_Olambda__570,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_jq(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_570
tff(fact_10134_ATP_Olambda__571,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_js(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_571
tff(fact_10135_ATP_Olambda__572,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_jt(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_572
tff(fact_10136_ATP_Olambda__573,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_ju(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_573
tff(fact_10137_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sp(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_574
tff(fact_10138_ATP_Olambda__575,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sr(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_575
tff(fact_10139_ATP_Olambda__576,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_so(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_576
tff(fact_10140_ATP_Olambda__577,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_sq(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu),aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_577
tff(fact_10141_ATP_Olambda__578,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mv(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(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_578
tff(fact_10142_ATP_Olambda__579,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(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_579
tff(fact_10143_ATP_Olambda__580,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mr(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mq(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_580
tff(fact_10144_ATP_Olambda__581,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_ws(Uu),Uua),Uub),Uuc) = times_3573771949741848930nteger(if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_Code_integer,zero_z3403309356797280102nteger)),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_581
tff(fact_10145_ATP_Olambda__582,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_wp(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_582
tff(fact_10146_ATP_Olambda__583,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wt(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(power_power_real(Uua),Uuc)) ).

% ATP.lambda_583
tff(fact_10147_ATP_Olambda__584,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_wr(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_584
tff(fact_10148_ATP_Olambda__585,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wq(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_585
tff(fact_10149_ATP_Olambda__586,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_qd(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_qc(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ir(Uu),Uub),Uuc))) ).

% ATP.lambda_586
tff(fact_10150_ATP_Olambda__587,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_qa(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_pz(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bp(Uu),Uub),Uuc))) ).

% ATP.lambda_587
tff(fact_10151_ATP_Olambda__588,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_int(aa_fun1384843155787029862at_int(aa_fun6066863168309489065at_int(aTP_Lamm_px(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_nat3598295560461965541BT_int(aTP_Lamm_pw(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_bo(Uu),Uub),Uuc))) ).

% ATP.lambda_588
tff(fact_10152_ATP_Olambda__589,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_qr(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_iq(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ir(Uu),Uub),Uuc))) ).

% ATP.lambda_589
tff(fact_10153_ATP_Olambda__590,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_qp(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_im(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bx(Uu),Uub),Uuc))) ).

% ATP.lambda_590
tff(fact_10154_ATP_Olambda__591,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_qn(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_ii(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bp(Uu),Uub),Uuc))) ).

% ATP.lambda_591
tff(fact_10155_ATP_Olambda__592,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_ql(Uu),Uua),Uub),Uuc) = groups6361806394783013919BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_ie(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_bo(Uu),Uub),Uuc))) ).

% ATP.lambda_592
tff(fact_10156_ATP_Olambda__593,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_qj(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_hz(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_ia(Uu),Uub),Uuc))) ).

% ATP.lambda_593
tff(fact_10157_ATP_Olambda__594,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_qh(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_hr(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_br(Uu),Uub),Uuc))) ).

% ATP.lambda_594
tff(fact_10158_ATP_Olambda__595,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: int] : aa_int_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_qf(Uu),Uua),Uub),Uuc) = groups6359315924273963643BT_int(aa_int1343382402272391561BT_int(aTP_Lamm_hn(Uua),Uuc),collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_595
tff(fact_10159_ATP_Olambda__596,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_ja(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_iz(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bt(Uu),Uub),Uuc))) ).

% ATP.lambda_596
tff(fact_10160_ATP_Olambda__597,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: complex] : aa_complex_complex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_iw(Uu),Uua),Uub),Uuc) = groups1794756597179926696omplex(aa_com2862003112617944841omplex(aTP_Lamm_iv(Uua),Uuc),collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_bs(Uu),Uub),Uuc))) ).

% ATP.lambda_597
tff(fact_10161_ATP_Olambda__598,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_is(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_iq(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ir(Uu),Uub),Uuc))) ).

% ATP.lambda_598
tff(fact_10162_ATP_Olambda__599,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_in(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_im(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_bx(Uu),Uub),Uuc))) ).

% ATP.lambda_599
tff(fact_10163_ATP_Olambda__600,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_ij(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_ii(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_bp(Uu),Uub),Uuc))) ).

% ATP.lambda_600
tff(fact_10164_ATP_Olambda__601,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_if(Uu),Uua),Uub),Uuc) = groups771621172384141258BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_ie(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_bo(Uu),Uub),Uuc))) ).

% ATP.lambda_601
tff(fact_10165_ATP_Olambda__602,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_ib(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_hz(Uua),Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_ia(Uu),Uub),Uuc))) ).

% ATP.lambda_602
tff(fact_10166_ATP_Olambda__603,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_hw(Uu),Uua),Uub),Uuc) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hv(Uua),Uuc),collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_bv(Uu),Uub),Uuc))) ).

% ATP.lambda_603
tff(fact_10167_ATP_Olambda__604,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_hs(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_hr(Uua),Uuc),collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_br(Uu),Uub),Uuc))) ).

% ATP.lambda_604
tff(fact_10168_ATP_Olambda__605,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: int] : aa_int_int(aa_fun5422038952152218214nt_int(aa_fun3638225402958239493nt_int(aTP_Lamm_ho(Uu),Uua),Uub),Uuc) = groups769130701875090982BT_int(aa_int1343382402272391561BT_int(aTP_Lamm_hn(Uua),Uuc),collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_bq(Uu),Uub),Uuc))) ).

% ATP.lambda_605
tff(fact_10169_ATP_Olambda__606,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_vz(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_vy(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_606
tff(fact_10170_ATP_Olambda__607,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_vw(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_vv(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_607
tff(fact_10171_ATP_Olambda__608,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_vu(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_vt(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_608
tff(fact_10172_ATP_Olambda__609,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_vs(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_vr(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_609
tff(fact_10173_ATP_Olambda__610,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_vq(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_vp(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_610
tff(fact_10174_ATP_Olambda__611,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_abs(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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_611
tff(fact_10175_ATP_Olambda__612,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_abq(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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_612
tff(fact_10176_ATP_Olambda__613,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_abp(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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_613
tff(fact_10177_ATP_Olambda__614,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_abo(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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_614
tff(fact_10178_ATP_Olambda__615,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nl(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_615
tff(fact_10179_ATP_Olambda__616,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nr(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(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(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_616
tff(fact_10180_ATP_Olambda__617,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nn(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_617
tff(fact_10181_ATP_Olambda__618,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_np(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_618
tff(fact_10182_ATP_Olambda__619,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aap(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_619
tff(fact_10183_ATP_Olambda__620,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mq(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_620
tff(fact_10184_ATP_Olambda__621,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_621
tff(fact_10185_ATP_Olambda__622,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ms(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_622
tff(fact_10186_ATP_Olambda__623,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yh(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(comm_s7457072308508201937r_real(Uu),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_623
tff(fact_10187_ATP_Olambda__624,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_yf(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_624
tff(fact_10188_ATP_Olambda__625,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_yg(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_625
tff(fact_10189_ATP_Olambda__626,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_yb(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_626
tff(fact_10190_ATP_Olambda__627,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ye(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(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_627
tff(fact_10191_ATP_Olambda__628,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_yc(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_628
tff(fact_10192_ATP_Olambda__629,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ya(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_629
tff(fact_10193_ATP_Olambda__630,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_yd(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_630
tff(fact_10194_ATP_Olambda__631,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nt(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_631
tff(fact_10195_ATP_Olambda__632,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(power_power_real(Uu),Uuc)) ).

% ATP.lambda_632
tff(fact_10196_ATP_Olambda__633,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nu(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_633
tff(fact_10197_ATP_Olambda__634,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nv(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_634
tff(fact_10198_ATP_Olambda__635,axiom,
    ! [Uu: set_complex,Uua: fun_VE2128610149277289041x_bool,Uub: vEBT_VEBT,Uuc: complex] :
      ( pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_it(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_635
tff(fact_10199_ATP_Olambda__636,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ix(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_636
tff(fact_10200_ATP_Olambda__637,axiom,
    ! [Uu: set_nat,Uua: fun_VE3763956218935105199t_bool,Uub: vEBT_VEBT,Uuc: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ic(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_637
tff(fact_10201_ATP_Olambda__638,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_io(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_638
tff(fact_10202_ATP_Olambda__639,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_ig(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_639
tff(fact_10203_ATP_Olambda__640,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_ik(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_640
tff(fact_10204_ATP_Olambda__641,axiom,
    ! [Uu: set_int,Uua: fun_VE3303673267288458579t_bool,Uub: vEBT_VEBT,Uuc: int] :
      ( pp(aa_int_bool(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_hl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_641
tff(fact_10205_ATP_Olambda__642,axiom,
    ! [Uu: set_int,Uua: fun_co2381122952212249071t_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hx(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_642
tff(fact_10206_ATP_Olambda__643,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_hp(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_643
tff(fact_10207_ATP_Olambda__644,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_ht(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_644
tff(fact_10208_ATP_Olambda__645,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2128610149277289041x_bool,Uub: complex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_bs(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_645
tff(fact_10209_ATP_Olambda__646,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3763956218935105199t_bool,Uub: nat,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_bo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_646
tff(fact_10210_ATP_Olambda__647,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3303673267288458579t_bool,Uub: int,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_bq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_647
tff(fact_10211_ATP_Olambda__648,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_ir(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_648
tff(fact_10212_ATP_Olambda__649,axiom,
    ! [Uu: set_complex,Uua: fun_co2381122952212249071t_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun2352441141717865852x_bool(aTP_Lamm_ia(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_649
tff(fact_10213_ATP_Olambda__650,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_bt(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_650
tff(fact_10214_ATP_Olambda__651,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_bp(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_651
tff(fact_10215_ATP_Olambda__652,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_br(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_652
tff(fact_10216_ATP_Olambda__653,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_bw(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_653
tff(fact_10217_ATP_Olambda__654,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_bu(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_654
tff(fact_10218_ATP_Olambda__655,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_bv(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_655
tff(fact_10219_ATP_Olambda__656,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_bx(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_656
tff(fact_10220_ATP_Olambda__657,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nm(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_657
tff(fact_10221_ATP_Olambda__658,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ns(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_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(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_658
tff(fact_10222_ATP_Olambda__659,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_no(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_659
tff(fact_10223_ATP_Olambda__660,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nq(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_660
tff(fact_10224_ATP_Olambda__661,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nx(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_661
tff(fact_10225_ATP_Olambda__662,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_oa(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uu),Uuc)),aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_662
tff(fact_10226_ATP_Olambda__663,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ny(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_663
tff(fact_10227_ATP_Olambda__664,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nz(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_664
tff(fact_10228_ATP_Olambda__665,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_xz(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_665
tff(fact_10229_ATP_Olambda__666,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_aau(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_666
tff(fact_10230_ATP_Olambda__667,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_adl(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_667
tff(fact_10231_ATP_Olambda__668,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_aas(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_668
tff(fact_10232_ATP_Olambda__669,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aaw(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_669
tff(fact_10233_ATP_Olambda__670,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aay(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_670
tff(fact_10234_ATP_Olambda__671,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_cy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_671
tff(fact_10235_ATP_Olambda__672,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_di(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_672
tff(fact_10236_ATP_Olambda__673,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_dg(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_673
tff(fact_10237_ATP_Olambda__674,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_dq(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_674
tff(fact_10238_ATP_Olambda__675,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_da(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_675
tff(fact_10239_ATP_Olambda__676,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_dk(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_676
tff(fact_10240_ATP_Olambda__677,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_dc(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_677
tff(fact_10241_ATP_Olambda__678,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_dm(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_678
tff(fact_10242_ATP_Olambda__679,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_de(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_679
tff(fact_10243_ATP_Olambda__680,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_do(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_680
tff(fact_10244_ATP_Olambda__681,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_ds(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_681
tff(fact_10245_ATP_Olambda__682,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_ec(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_682
tff(fact_10246_ATP_Olambda__683,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_ea(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_683
tff(fact_10247_ATP_Olambda__684,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_ek(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_684
tff(fact_10248_ATP_Olambda__685,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_du(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_685
tff(fact_10249_ATP_Olambda__686,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_ee(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_686
tff(fact_10250_ATP_Olambda__687,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_dw(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_687
tff(fact_10251_ATP_Olambda__688,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_eg(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_688
tff(fact_10252_ATP_Olambda__689,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_dy(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_689
tff(fact_10253_ATP_Olambda__690,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_ei(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_690
tff(fact_10254_ATP_Olambda__691,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_um(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_691
tff(fact_10255_ATP_Olambda__692,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_un(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_692
tff(fact_10256_ATP_Olambda__693,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tk(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_693
tff(fact_10257_ATP_Olambda__694,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_694
tff(fact_10258_ATP_Olambda__695,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tn(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(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,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_695
tff(fact_10259_ATP_Olambda__696,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tq(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_696
tff(fact_10260_ATP_Olambda__697,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_vp(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_697
tff(fact_10261_ATP_Olambda__698,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_vv(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_698
tff(fact_10262_ATP_Olambda__699,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_vr(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_699
tff(fact_10263_ATP_Olambda__700,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_vy(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_700
tff(fact_10264_ATP_Olambda__701,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_vt(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_701
tff(fact_10265_ATP_Olambda__702,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_qq(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_io(Uu),Uub),Uuc))) ).

% ATP.lambda_702
tff(fact_10266_ATP_Olambda__703,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_qb(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_io(Uu),Uub),Uuc))) ).

% ATP.lambda_703
tff(fact_10267_ATP_Olambda__704,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_qo(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_ik(Uu),Uub),Uuc))) ).

% ATP.lambda_704
tff(fact_10268_ATP_Olambda__705,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_qm(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_ig(Uu),Uub),Uuc))) ).

% ATP.lambda_705
tff(fact_10269_ATP_Olambda__706,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_py(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_ig(Uu),Uub),Uuc))) ).

% ATP.lambda_706
tff(fact_10270_ATP_Olambda__707,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_qk(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ic(Uu),Uub),Uuc))) ).

% ATP.lambda_707
tff(fact_10271_ATP_Olambda__708,axiom,
    ! [Uu: set_nat,Uua: fun_VE7331017413731640256at_int,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun6141810409722670662BT_int(aa_fun6474440919679335769BT_int(aTP_Lamm_pv(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_VEB6167613734120275313at_int(Uua,Uuc),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ic(Uu),Uub),Uuc))) ).

% ATP.lambda_708
tff(fact_10272_ATP_Olambda__709,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_qi(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hx(Uu),Uub),Uuc))) ).

% ATP.lambda_709
tff(fact_10273_ATP_Olambda__710,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_qg(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_hp(Uu),Uub),Uuc))) ).

% ATP.lambda_710
tff(fact_10274_ATP_Olambda__711,axiom,
    ! [Uu: set_int,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_qe(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_VEB7166967446021329869nt_int(Uua,Uuc),collect_int(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_hl(Uu),Uub),Uuc))) ).

% ATP.lambda_711
tff(fact_10275_ATP_Olambda__712,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_iy(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ix(Uu),Uub),Uuc))) ).

% ATP.lambda_712
tff(fact_10276_ATP_Olambda__713,axiom,
    ! [Uu: set_complex,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_iu(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_VEB5608662635949521omplex(Uua,Uuc),collect_complex(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_it(Uu),Uub),Uuc))) ).

% ATP.lambda_713
tff(fact_10277_ATP_Olambda__714,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_ip(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_io(Uu),Uub),Uuc))) ).

% ATP.lambda_714
tff(fact_10278_ATP_Olambda__715,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_il(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_ik(Uu),Uub),Uuc))) ).

% ATP.lambda_715
tff(fact_10279_ATP_Olambda__716,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_ih(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_ig(Uu),Uub),Uuc))) ).

% ATP.lambda_716
tff(fact_10280_ATP_Olambda__717,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_id(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_ic(Uu),Uub),Uuc))) ).

% ATP.lambda_717
tff(fact_10281_ATP_Olambda__718,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_hy(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun8206557716905444476t_bool(aTP_Lamm_hx(Uu),Uub),Uuc))) ).

% ATP.lambda_718
tff(fact_10282_ATP_Olambda__719,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_hu(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_ht(Uu),Uub),Uuc))) ).

% ATP.lambda_719
tff(fact_10283_ATP_Olambda__720,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_hq(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_hp(Uu),Uub),Uuc))) ).

% ATP.lambda_720
tff(fact_10284_ATP_Olambda__721,axiom,
    ! [Uu: set_int,Uua: fun_VE294566951569886236nt_int,Uub: fun_VE3303673267288458579t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun4615758601980977314BT_int(aa_fun627869447348354137BT_int(aTP_Lamm_hm(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_VEB7166967446021329869nt_int(Uua,Uuc),collect_int(aa_VEB1707969225601237452t_bool(aa_fun978093584635342820t_bool(aTP_Lamm_hl(Uu),Uub),Uuc))) ).

% ATP.lambda_721
tff(fact_10285_ATP_Olambda__722,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_le(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_722
tff(fact_10286_ATP_Olambda__723,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ld(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_723
tff(fact_10287_ATP_Olambda__724,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_si(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_724
tff(fact_10288_ATP_Olambda__725,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_xb(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_xa(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_725
tff(fact_10289_ATP_Olambda__726,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_wz(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_wy(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_726
tff(fact_10290_ATP_Olambda__727,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_wx(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_ww(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_727
tff(fact_10291_ATP_Olambda__728,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_wv(Uu),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_wu(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_728
tff(fact_10292_ATP_Olambda__729,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_abr(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_abq(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(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_729
tff(fact_10293_ATP_Olambda__730,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_tu(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tt(Uua),Uuc),Uud)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_real(power_power_real(Uub),Uud)) ).

% ATP.lambda_730
tff(fact_10294_ATP_Olambda__731,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_tr(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tq(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_731
tff(fact_10295_ATP_Olambda__732,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_to(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tn(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_732
tff(fact_10296_ATP_Olambda__733,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_tl(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tk(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_733
tff(fact_10297_ATP_Olambda__734,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zs(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_734
tff(fact_10298_ATP_Olambda__735,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zu(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(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_735
tff(fact_10299_ATP_Olambda__736,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zt(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_736
tff(fact_10300_ATP_Olambda__737,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zh(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_737
tff(fact_10301_ATP_Olambda__738,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zl(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(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_738
tff(fact_10302_ATP_Olambda__739,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zj(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_739
tff(fact_10303_ATP_Olambda__740,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zi(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_740
tff(fact_10304_ATP_Olambda__741,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zm(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(gbinomial_real(aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_741
tff(fact_10305_ATP_Olambda__742,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_zk(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,Uua)),Uud)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_742
tff(fact_10306_ATP_Olambda__743,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_wu(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_743
tff(fact_10307_ATP_Olambda__744,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_xa(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(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_744
tff(fact_10308_ATP_Olambda__745,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_ww(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_745
tff(fact_10309_ATP_Olambda__746,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_wy(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_746
tff(fact_10310_ATP_Olambda__747,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_aco(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)))
        & ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),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,I5)),X_13))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I5)) ) )
        & ( ( Uuc = Uud )
         => ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),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))
            & ! [X3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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),X3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X3))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_747
tff(fact_10311_ATP_Olambda__748,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_tx(Uu),Uua) = Uu ).

% ATP.lambda_748
tff(fact_10312_ATP_Olambda__749,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ty(Uu),Uua) = Uu ).

% ATP.lambda_749
tff(fact_10313_ATP_Olambda__750,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cd,Uu) = Uu ).

% ATP.lambda_750
tff(fact_10314_ATP_Olambda__751,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ce,Uu) = Uu ).

% ATP.lambda_751
tff(fact_10315_ATP_Olambda__752,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cf,Uu) = Uu ).

% ATP.lambda_752
tff(fact_10316_ATP_Olambda__753,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cg,Uu) = Uu ).

% ATP.lambda_753
tff(fact_10317_ATP_Olambda__754,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ch,Uu) = Uu ).

% ATP.lambda_754
tff(fact_10318_ATP_Olambda__755,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_by,Uu) = zero_zero_complex ).

% ATP.lambda_755
tff(fact_10319_ATP_Olambda__756,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_bz,Uu) = zero_zero_real ).

% ATP.lambda_756
tff(fact_10320_ATP_Olambda__757,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ca,Uu) = zero_zero_rat ).

% ATP.lambda_757
tff(fact_10321_ATP_Olambda__758,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_oi,Uu) = zero_zero_complex ).

% ATP.lambda_758
tff(fact_10322_ATP_Olambda__759,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_fi,Uu) = zero_zero_real ).

% ATP.lambda_759
tff(fact_10323_ATP_Olambda__760,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cb,Uu) = zero_zero_nat ).

% ATP.lambda_760
tff(fact_10324_ATP_Olambda__761,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_oj,Uu) = zero_zero_int ).

% ATP.lambda_761
tff(fact_10325_ATP_Olambda__762,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cc,Uu) = zero_zero_int ).

% ATP.lambda_762
tff(fact_10326_ATP_Olambda__763,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_bl,Uu))
    <=> $false ) ).

% ATP.lambda_763
tff(fact_10327_ATP_Olambda__764,axiom,
    ! [Uu: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_bj,Uu))
    <=> $false ) ).

% ATP.lambda_764
tff(fact_10328_ATP_Olambda__765,axiom,
    ! [Uu: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_bk,Uu))
    <=> $false ) ).

% ATP.lambda_765
tff(fact_10329_ATP_Olambda__766,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_bm,Uu))
    <=> $false ) ).

% ATP.lambda_766
tff(fact_10330_ATP_Olambda__767,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_bn,Uu))
    <=> $false ) ).

% ATP.lambda_767
tff(fact_10331_ATP_Olambda__768,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_aal,Uu))
    <=> $true ) ).

% ATP.lambda_768

% Helper facts (66)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(fconj(P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fdisj(P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y: int] : aa_int_int(if_int(fFalse,X2),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y: int] : aa_int_int(if_int(fTrue,X2),Y) = X2 ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X2),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X2),Y) = X2 ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y: rat] : aa_rat_rat(if_rat(fFalse,X2),Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y: rat] : aa_rat_rat(if_rat(fTrue,X2),Y) = X2 ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X2: nat] :
      ( ~ pp(aa_nat_bool(P,X2))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X2),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X2),Y) = X2 ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X2: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y: rat] :
      ( ( X2 != Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(fequal_rat,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X2: bool,Y: bool] :
      ( ( X2 != Y )
      | pp(aa_bool_bool(fequal_bool(X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X2: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(fequal_bool(X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_bool] : aa_real_bool(P,fChoice_real(P)) = aa_fun3352386287069783311l_bool(fEx_real,P) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X2),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X2),Y) = X2 ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y: complex] :
      ( ( X2 != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y: extended_enat] : if_Extended_enat(fFalse,X2,Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y: extended_enat] : if_Extended_enat(fTrue,X2,Y) = X2 ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y: code_integer] : if_Code_integer(fFalse,X2,Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y: code_integer] : if_Code_integer(fTrue,X2,Y) = X2 ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fFalse,X2),Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fTrue,X2),Y) = X2 ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_VEBT_VEBT_bool,X2: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P))
      | pp(aa_VEBT_VEBT_bool(P,X2)) ) ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y: code_integer] :
      ( ( X2 != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y: set_nat] :
      ( ( X2 != Y )
      | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X2 != Y )
      | pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,X2),Y))
      | ( X2 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y: list_nat] :
      ( ( X2 != Y )
      | pp(aa_list_nat_bool(fequal_list_nat(X2),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y: list_nat] :
      ( ~ pp(aa_list_nat_bool(fequal_list_nat(X2),Y))
      | ( X2 = Y ) ) ).

tff(help_If_2_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: fun_int_int,Y: fun_int_int] : if_fun_int_int(fFalse,X2,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: fun_int_int,Y: fun_int_int] : if_fun_int_int(fTrue,X2,Y) = X2 ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X2,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X2,Y) = X2 ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fFalse,X2,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fTrue,X2,Y) = X2 ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R),aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X2,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X2,Y) = X2 ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R) = aa_fun_nat_bool_bool(P,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,
    ! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R) = aa_boo8970175072276269406l_bool(P,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,
    ! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R),Q) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ? [Y3: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))),Y3)) ).

%------------------------------------------------------------------------------